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6/01 cJClRC/DanDuo.p65-p.15

UNDERSTANDING ALGEBRA AND FUNCTIONS:
AN EXPLORATION OF THE LEARNING EXPERIENCES
OF PREVIOUSLY UNSUCCESSFUL STUDENTS
IN CORE-PLUS COURSE 1A

By

Angia E. Sperfslage Macomber

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Department of Teacher Education

2003

ABSTRACT
UNDERSTANDING ALGEBRA AND FUNCTIONS:
AN EXPLORATION OF THE LEARNING EXPERIENCES
OF PREVIOUSLY UNSUCCESSFUL STUDENTS
IN CORE-PLUS COURSE 1A
By

Angia E. Sperfslage Macomber

A major focus of mathematics reform at the secondary level has been directed at
the teaching and learning of algebra, driven by the belief that mathematics
instruction—and algebra instruction, in particular—must effectively reach
all students. It is widely agreed upon in mathematics reform literature that
algebra is a gatekeeping course whereby students who do not succeed are denied
access to equal participation in a technologically oriented society; moreover,
algebra is accessible to all students only when it can be understood
conceptually, explored within contexts that are meaningful to students.

In this study, the author explores the learning experiences—focusing on the
algebra and functions content strand—of six students, each of whom had previously
experienced non-success in school mathematics, in the first part of the first course of
Core-Plus, a reform high-school mathematics curriculum. Through an analysis of
classroom observations and in-depth interviews conducted at three different moments in
time of the course, the author investigates the nature of the students’ previous non-
successes, their experiences of the Core-Plus curriculum, and their understandings of

algebra and functions as presented in Core-Plus Course 1A. Afier developing individual

portraits of each student, the author reads across all six participants for common features

and patterns of their school-mathematics histories, their experiences of the Core-Plus
curriculum, and their understandings of algebra and functions.

The findings coalesce around issues attendant to representations of mathematical
phenomena, the role of contextualized situations in providing access to abstract
mathematics, and a construct called “mathematics community membership” that captures
a number of qualities that the author claims to be closely associated with navigating a
reform curriculum successfully. The implications call for both high school mathematics
teachers and mathematics teacher educators to devote serious attention to students’
alienation from the subject and to explicitly socialize them into expressly mathematical

modes of thinking, speaking, and behaving.

C0pyright by
ANGIA E. SPERFSLAGE MACOMBER
2003

TO Kirsten
who challenged me, as a teacher,
to make Algebra a meaningful learning experience
for previously unsuccessful students

ACKNOWLEDGEMENTS

TO me, this is the most important part of my dissertation. I have waited to
compose it until the very end; it is the finishing touch and the culminating piece of this
work. I recognize that this project is not the work of just the one person whose name
appears in the byline; it is the collaboration of many people to whom I am deeply
indebted. Without them, this volume would never have come together. More
importantly, without them, I would not have the benefit of knowing and understanding all
that I have learned throughout this course of study and this research project.

To the young women who were the focus of my study, I am very grateful. I
learned a great deal through my interactions with and observations of them. Through
their participation, they have contributed a great deal to the betterment of mathematics
education and, because of that, I credit each one of them with being highly successful!

To the math teachers at the high school where I conducted my study, I am also
very grateful. They welcomed me as one of their own, and I learned a great deal from
them as well, visiting their classes and listening to their lunchtime conversations. Their
openness and collegial spirit allowed me to see that, through their collective commitment
to effectively reaching all students through their mathematics instruction, they too have
contributed a great deal to the betterment of mathematics education.

To (Mr) Bill Rosenthal, my dissertation director, I don’t have the words to
express the depth of my appreciation. We began and ended our time at Michigan State
University simultaneously (1992-1998), yet our greatest work of collaboration had not
yet begun. If it weren’t for Bill, I am convinced this project would never have

commenced; and I am absolutely certain that I would not have finished what was started

vi

so many years earlier were it not for him. He believed in me more than I did myself. Bill
encouraged, supported, empathized with, listened to, understood, challenged, and pushed
me all along the way, from the time I enrolled in his Mathematical Ways of Knowing
course in 1994 to the final editing of this dissertation. I can only hope to be able to
contribute as much to my own students’ pedagogical betterment as Bill has contributed to
mine.

To the members of my dissertation committee—Sandy Wilcox, Susan Melnick,
Jack Smith, and Dan Chazan—I also extend my appreciation. In addition to their
valuable contributions as members of my committee, they have contributed a great deal
to my growth as a learner, a teacher, and a scholar. I had the privilege of learning most of
what I know about qualitative research methods from Sandy. My first foray into teaching
elementary math methods was in collaboration with her; and I finished my graduate
assistant career at MSU working on a research project that Sandy was directing. I
appreciate Sandy most for her mentorship and for the serious approach she takes when
considering the fledgling research of her students. I had the privilege of getting to know
Susan during my first year at MSU in the professional seminar course she co-taught. I
appreciate Susan most of all for her frankness, her tough-mindedness, and her sense of
humor. Susan has “kept me honest” when it comes to addressing issues of diverse
learners and equity in education. While I did not have the privilege of taking a course
from either Jack or Dan, I have learned a tremendous amount from them about
mathematics and mathematics education. I have benefited, not only from the research
about which they have written, but most of all from the many hours they have spent

sharing their research and contributing ideas to the research of others in Math Education

vii

Seminars and in the Math Learning Research Group. I greatly appreciate Jack and Dan
for the encouragement and support they give to graduate students who are still trying to
find their way doing math education research.

To Taylor University where I am currently employed, I am greatly beholden.
Through their Doctoral Studies Assistance program, I was granted a year’s leave with
funding in order to complete my degree. I was provided with a research office on
campus where I was able to work without distraction while I collected and analyzed my
data. My colleagues at Taylor have been tremendously encouraging and supportive,
especially during this second year of work on my dissertation as I have split my time
between my teaching responsibilities and writing. My good friend and mentor, Pam
Medows deserves special credit for standing in the gap for me—doing things such as
advising my students and grading papers—so that I could devote as much time as
possible to finishing my dissertation. I have been tremendously blessed to be a part of
the Taylor community whose support has been a large contributing factor to my
completing this project.

To my mentors and fi'iends at Michigan State University, I also owe a debt of
gratitude. So many contributed to my education there, in ways too numerous to mention,
yet I would be remiss if I did not acknowledge Glenda Lappan, Deborah Ball, Maggie
Lampert, and Deb Smith for the ways in which they helped me grow as a learner, as a
teacher, and as a scholar. I also wish to acknowledge fellow graduate students Kara
Suzuka, Sarah Theule Lubienski, Ruth Heaton, and Duane Castanier for the fiiendship
they shared with me and for the significant ways in which they, too, helped me grow as a

learner, a teacher, and a scholar.

viii

To my parents who have always loved me and supported me, I am so very
indebted. They have always wanted the best for me and have done whatever they could
to provide that for me. From the time they saw me off to kindergarten to the completion
of this “paper,” they have encouraged me to fulfill my potential through education. I
could not have reached this point without them.

To my husband Phil, I am indebted most of all. Mere words cannot adequately
express my love and appreciation for all that he has done to help me finish my degree.
Phil has not known me apart from having a dissertation to write, and he has patiently
endured the process without complaint as I have devoted much more time and energy to
the completion of my dissertation than I have to him. He has done far more than his
share of laundry, cooking, cleaning, doing dishes, running errands, and getting us moved
into our “new” house during Spring Break of this year! And throughout this long
journey, Phil has not wavered in providing unselfish support, offering encouragement,
sharing laughter, lending a listening ear, and showing unconditional love. God has
blessed me in indescribable ways by bringing Phil into my life. Without him, I never

would have become “Dr. Macomber.”

TABLE OF CONTENTS

LIST OF TABLES ........................................................................................................... xiv
LIST OF FIGURES ....................................................................................................... xviii
CHAPTER 1
INTRODUCTION .............................................................................................................. 1
My Teaching Experiences .............................................................................................. 1
Graduate School ............................................................................................................. 3
Back to Teaching ........................................................................................................... 5
My Consequent Research Interests ................................................................................ 7
My Questions ............................................................................................................... 10
CHAPTER 2
OVERVIEW OF MATHEMATICS-EDUCATION AND ALGEBRA REFORM .......... 12
Everybody Counts ........................................................................................................ 12
Algebra for All ............................................................................................................. l4
Reform-based secondary mathematics curriculum and the Core-Plus Mathematics
Project .......................................................................................................................... 16
The “math wars” .......................................................................................................... 19
Further research needed ............................................................................................... 21
Occupying a unique spot in the existing literature ....................................................... 22
CHAPTER 3
CONCEPTUAL FRAMEWORK ...................................................................................... 25
Issues related to math and algebra reform ................................................................... 25
Issues pertaining to “previously unsuccessful” students ............................................. 26
Characteristics to be determined from students’ self-reports .......................... 27
Characteristics to be determined from personal interviews and
observations ..................................................................................................... 28
A map of the algebra terrain ........................................................................................ 30
Issues not addressed in this study ................................................................................ 32
CHAPTER 4
METHODS OF DATA COLLECTION AND ANALYSIS ............................................. 35
Research Context ......................................................................................................... 35
Selecting a site for my research ....................................................................... 35
The Community ............................................................................................... 36
The School ....................................................................................................... 37
The Curriculum ................................................................................................ 38
The Students ..................................................................................................... 39
Data Collection ........................................................................................................... 40
Initial Survey .................................................................................................... 4O

Classroom Observations .................................................................................. 43

Interviews ......................................................................................................... 46

Data Analysis ............................................................................................................... 48

Analysis of Initial Survey Data during the Data Collection Phase .................. 48

Analysis of Interview Data during the Data Collection Phase ........................ 50

Further Analysis of Initial Survey and Interview Data .................................... 52

Writing as Data Analysis ............................................................................................. 54

Portraits of the Six Participants ........................................................................ 55

Use of the Algebra Map ................................................................................... 55

The Emergence of Findings ............................................................................. 56
CHAPTER 5

PORTRAITS OF THE SIX PARTICIPANTS .................................................................. 58

Organization of the Portraits ........................................................................................ 59

Adrienne ....................................................................................................................... 6O

Distinguishing Characteristics of Adrienne ..................................................... 60

Adrienne’s Previous Non-Success in School Mathematics ............................. 62

Personal Math History ............................................................................... 62

Entering Attitudes Towards Mathematics ................................................. 63

Adrienne’s Experiences of the Core-Plus Course 1A Curriculum .................. 65

Adrienne’s Performance on Algebra and Functions Tasks .............................. 69

Sara .............................................................................................................................. 71

Distinguishing Characteristics of Sara ............................................................. 71

Sara’s Previous Non-Success in School Mathematics ..................................... 73

Personal Math History ............................................................................... 73

Entering Attitudes Towards Mathematics ................................................. 73

Sara’s Experiences of the Core-Plus Course 1A Curriculum .......................... 75

Sara’s Performance on Algebra and Functions Tasks ..................................... 79

Amy .............................................................................................................................. 8 1

Distinguishing Characteristics of Amy ............................................................ 81

Amy’s Previous Non-Success in School Mathematics .................................... 81

Personal Math History ............................................................................... 81

Entering Attitudes Towards Mathematics ................................................. 83

Amy’s Experiences of the Core-Plus Course 1A Curriculum ......................... 86

Amy’s Performance on Algebra and Functions Tasks .................................... 88

Joy ................................................................................................................................ 9O

Distinguishing Characteristics of Joy .............................................................. 90

Joy’s Previous Non-Success in School Mathematics ...................................... 91

Personal Math History ............................................................................... 91

Entering Attitudes Towards Mathematics ................................................. 91

Joy’s Experiences of the Core-Plus Course 1A Curriculum ............................ 94

Joy’s Performance on Algebra and Functions Tasks ....................................... 97

Terri ............................................................................................................................ 100

Distinguishing Characteristics of Terri .......................................................... 100

Terri’s Previous Non-Success in School Mathematics .................................. 101

Personal Math History ............................................................................. 101

xi

Entering Attitudes Towards Mathematics ............................................... 101

Terri’s Experiences of the Core-Plus Course 1A Curriculum ....................... 102
Terri’s Performance on Algebra and Functions Tasks .................................. 104
Monica ....................................................................................................................... 107
Distinguishing Characteristics of Monica ...................................................... 107
Monica’s Previous Non-Success in School Mathematics .............................. 108
Personal Math History ............................................................................. 108
Entering Attitudes Towards Mathematics ............................................... 108
Monica’s Experiences of the Core-Plus Course 1A Curriculum ................... 109
Monica’s Performance on Algebra and Functions Tasks .............................. 111
CHAPTER 6
BROADER DATA ANALYSIS AND INTERPRETATIONS ....................................... 114
Patterns across Participants ....................................................................................... 115
Characteristics of these students’ previous non-successes in school
mathematics ................................................................................................... 1 15
Characteristics of these students’ learning experiences in school
mathematics .................................................................................................. 1 l8
Characteristics of these students’ understandings of algebra and
functions ......................................................................................................... 126
Specific areas of the algebra terrain ......................................................... 132
Various Representations ................................................................. 132
Means of Solution ........................................................................... 135
Levels of Abstraction ...................................................................... 142
Functions ......................................................................................... 146
Number Relationships ..................................................................... 151
Number Groups ............................................................................... 153
Commonalities Of 9articipants’ understandings of algebra and
functions ................................................................................................... 15 3
Initiating and Connecting Representations ..................................... 153
Fragility of Understanding .............................................................. 156
Interpretations ............................................................................................................ 158
Contextualized Versus Abstracted Mathematics .......................................... 159
Language Issues ............................................................................................. 162
“Mathematics Community Membership” ...................................................... 164
CHAPTER 7
IMPLICATIONS
A Letter to High School Math Teachers .................................................................... 167
A Letter to Math Teacher Educators .......................................................................... 173
APPENDIX A
DATA COLLECTION INSTRUMENTS ....................................................................... 179

xii

APPENDIX B
INSTRUMENTS OF ANALYSIS, DATA SUMMARIES, AND
LONGER DATA EXAMPLES ....................................................................................... 202

BIBLIOGRAPHY ............................................................................................................ 250

xiii

LIST OF TABLES

Table 3.1 — Map of the Algebra Terrain ............................................................................ 33
Table 4.1 — Summary data on interview participants as derived from initial survey ........ 43
Table 4.2 — Summary of classroom observations of interview cohort .............................. 44
Table 4.3 — Summary of interviewees’ math class enrollment .......................................... 44
Table 4.4 — Example of scoring procedure for survey items ............................................. 50
Table 4.5 — Performance Skills and their Types for Interview 1 Task 3 ........................... 53
Table 4.6— Definitions for Performance Skill Types ......................................................... 54
Table 5.1 — Adrienne’s scores on initial survey attitude questions by category ................ 64
Table 5.2 — Adrienne’s Familiarity Checklists .................................................................. 65

Table 5.3 — Adrienne’s responses to attitude questionnaires from interview sessions ...... 68

Table 5.4 -— Adrienne’s scores on the interview problem-solving tasks ............................ 69
Table 5.5 — Scoring Rubric for interview problem-solving tasks ...................................... 69
Table 5.6 — Adrienne’s performance means for evaluation items ..................................... 70
Table 5.7 -— Sara’s scores on initial survey attitude questions by category ........................ 75
Table 5.8 — Sara’s Familiarity Checklists .......................................................................... 78
Table 5.9 — Sara’s responses to attitude questionnaires from interview sessions .............. 78
Table 5.10 — Sara’s Scores on the interview problem-solving tasks ................................. 79
Table 5.11 — Sara’s performance means for evaluation items ........................................... 80

xiv

Table 5.12 — Amy’s Familiarity Checklists ....................................................................... 82

Table 5.13 — Amy’s scores on initial attitude questions by category ................................ 84
Table 5.14 — Amy’s responses to the initial survey attitude questions .............................. 85
Table 5.15 —- Amy’s responses to attitude questionnaires from interview sessions ........... 87
Table 5.16 — Amy’s scores on the interview problem-solving tasks ................................. 88
Table 5.17 — Amy’s performance means for evaluation items .......................................... 89
Table 5.18 — Joy’s responses to the initial survey attitude questions ................................ 92
Table 5.19 —— Joy’s scores on initial survey attitude questions by category ....................... 93
Table 5.20 —— Joy’s familiarity checklist for Chapter 1 topics ............................................ 95
Table 5.21 — Joy’s responses to attitude questionnaires from interview sessions ............. 97
Table 5.22 - Joy’s scores on the interview problem-solving tasks .................................... 98
Table 5.23 — Joy’s familiarity checklist for Chapters 2 and 3 ........................................... 99
Table 5.24 — Joy’s performance means for evaluation items ........................................... 100
Table 5.25 - Terri’s scores on initial survey attitude questions by category ................... 102
Table 5.26 — Terri’s responses to attitude questionnaires from interview sessions ......... 103
Table 5.27 — Terri’s scores on the interview problem-solving tasks ............................... 104
Table 5.28 — Terri’s familiarity checklist for Chapter 3 .................................................. 105
Table 5.29 - Terri’s performance means for evaluation items ........................................ 106
Table 5.30 — Monica’s scores on initial survey attitude questions by category .............. 109

Table 5.31 — Monica’s responses to attitude questionnaires from interview sessions ..... 1 10

XV

Table 5.32 — Monica’s scores on the interview problem-solving tasks ........................... 112
Table 5.33 — Monica’s performance means for evaluation items .................................... 113
Table 6.1 — Initial Survey Attitude Questions: Analysis of Participants’ Responses ...... 116
Table 6.2 — Core-Plus Course 1 Chapters 1,2,3 and their algebra content ...................... 128
Table 6.3 — Interview Tasks and their algebra content .................................................... 130
Table 6.4 — Four types of equations appearing in the interview problem-solving tasks .136
Table 6.5 — Performance scores for participants’ using rules to graph functions ............ 138
Table 6.6 — Levels of Abstraction encountered in the interview problem-solving tasks. 142
Table 6.7 — Evaluation Inventory for Levels of Abstraction ........................................... 144
Table 6.8 — Levels of Abstraction for Evaluation Items related to Linear Functions ...... 148
Table 6.9 — Evaluation Items related to Non-Linear Functions ....................................... 150
Table 6.10 - Evaluation Items related to Number Relationships .................................... 152

Table 6.1 - Participants’ holisitic performance scores on tasks with varying

Degrees of structure for problem solving ........................................................................ 155
Table A] — Summary of Instruments .............................................................................. 179
Table B.1 — Initial Survey Data Summary ....................................................................... 204
Table 8.2 — Explanation of Evaluation Item Types ......................................................... 206
Table B.3 — Interview One Tasks: Listing of Evaluation Items and Item Types ............. 207
Table B.4 - Interview Two Tasks: Listing of Evaluation Items and Item Types ............ 208
Table B.5 — Interview Three Tasks: Listing of Evaluation Items and Item Types .......... 209
Table B.6 — Task Evaluation Mean Score Summary ....................................................... 210

xvi

Table B.7 -- Evaluation Inventory for Various Representations ...................................... 213

Table B.8 — Evaluation Inventory for Means of Solution ................................................ 215
Table B.9 — Evaluation Inventory for Levels of Abstraction ........................................... 217
Table B. 10 — Evaluation Inventory for Functions ........................................................... 218
Table B.11 — Evaluation Inventory for Number Relationships ....................................... 219

xvii

LIST OF FIGURES

Figure 1.1 — First iteration of research focus ....................................................................... 8
Figure 1.2 — Second iteration of research focus ................................................................... 8
Figure 1.3 - Third iteration of research focus ...................................................................... 9
Figure 1.4 — Final iteration of research focus .................................................................... 10

Figure 5.1 — Histogram & Overlaid Boxplot of mean scores on initial survey attitude

questions, showing Sara’s placement ................................................................................ 74
Figure 6.1 — Monica’s work on Task 2 of Interview Two ............................................... 122
Figure 6.2 — Suggested solutions from Coxford et al. (1998a, p.66) ............................... 122
Figure 6.3 — Adrienne’s work on Task 3 of Interview Two ............................................ 137

Figure 6.4 — One student’s use of a systematic method to find all possible solutions
to the contextualized problem from the initial survey ..................................................... 165

Figure 6.5 — Another student’s less successful approach to the contextualized problem
from the initial survey ...................................................................................................... 165

Figure 8.1 — Correct circle diagram ................................................................................ 248

xviii

 

CHAPTER 1

INTRODUCTION

My Teaching Experiences

In Fall 1989, just months after the publication of NCTM’s original Standards
document, I enrolled in a secondary mathematics methods course at the University of
Kansas under the instruction of a soon-to-be member of the NCTM Board of Directors,
Dr. Beverly Nichols. Having caught the vision of math reform in my methods class, I
began my first job as a high school math teacher determined to make mathematics
learning a meaningful experience for my students. I taught in a rural school district in
northeast Kansas. The average graduating class was 50; the average class size was no
more than 20. I taught Pre-Algebra, Algebra Two, and Geometry the first year, Algebra
One, Algebra Two, and Geometry the second year. The algebra texts I used were the
classic “Dolciani” textbooks (Dolciani, brown, & Cole, 1984; Dolciani, Sorgenfrey,
Brown, & Kane, 1986).

Not having had a student-teaching experience in mathematics,‘ my first year as a
math teacher was eye opening. My algebra classes posed the greatest challenges. My
Pre-Algebra students (placed in this course, rather than Algebra One, because they were
thought to need remediation of basic math skills in order to prepare them to succeed in
Algebra One) were sick and tired of doing the same things year after year (Flanders,
1987). They complained that they had done “all that stuff” in the Pre-Algebra book

before. It was clear they felt stigmatized by having been placed in Pre—Algebra in the

 

IMathematics is an endorsement area that I added to my original teaching certificate. 1 student taught in
1987 when I first obtained my teaching certificate in French, but I was not required to do another student
teaching experience when adding the endorsement in math.

first place, and many lacked the belief that they were capable of learning math at all.
Many did not aspire to go to college, nor did their parents encourage them to do so. Math
was a necessary evil they had to endure to graduate high school. Many times they asked,
“Why do we need to know this?” And I struggled to help them, spending hours upon
hours trying to find real-life applications for the material they were learning, trying to
create projects or unique activities that would allow them to use their math skills in a
context that was interesting and perhaps meaningful to them. But this work was difficult,
time-consuming, and not often successful.

Then there were my Algebra Two classes. They really struggled, too! It was as if
all of their algebra learning had completely evaporated during the previous year when
they had taken Geometry. We limped along through the text, until I finally decided—in
January—that most of students’ difficulties stemmed from not being able to remember
the basic algorithms for algebraic/symbolic manipulation. Without these, the further we
progressed through the text (topics), the more difficult it became for them to be
successful. After spending a few weeks on rudimentary review of the rules for solving
algebraic equations, the highest-achieving students seemed to have been helped and
enjoyed more success than before. Yet the majority still struggled with, actually resisted,
the things we continued to work on, lamenting that they could memorize the steps well
enough, but Why did they do them? What sense did it make? How could they
understand what they were doing, instead of just blindly following the algorithms I made
them write down? Again, I struggled to help them, spending hours upon hours trying to
find real-life applications (from other textbooks) for the rules and procedures I was

teaching, trying to create projects or unique activities that might help them make

connections across mathematics topics. But this was difficult, time-consuming, and not
often successful. It soon became clear to me that, without a reform-minded curriculum,

teaching in reform-consistent ways proved extremely difficult indeed.

Graduate School

After two years of teaching high school in Kansas, I began my doctoral program
at Michigan State. My first assistantship involved working for Glenda Lappan, Bill
Fitzgerald, and Betty Phillips on the Connected Mathematics Project (CMP) middle
school curriculum. As I saw it, the major goal of this curriculum was to engage students
in investigations of math concepts in contexts that were meaningful, as opposed to the
“naked numbers” problems that were so common in traditional math curricula. I was
especially intrigued to make the acquaintance of Jim Fey, one of the five collaborating
authors of the CMP curriculum. Fey’s area of expertise was algebra and functions. He
was in charge of finding ways to introduce algebra and functions concepts via meaningful
contexts. At times he admitted he was truly challenged. He was determined nonetheless.

I also learned that Fey was a member of a high school mathematics curriculum-
writing team, the Core-Plus Mathematics Project (CPMP).2 The name Core-Plus, as I
understood it, was an outgrowth of the Standards concept of a core curriculum for high
school mathematics, which advocated that all students learn math in ways that connected
many topics each year (algebra, geometry, data analysis, probability, discrete math,
trigonometry, and so on) and delved more deeply into these interrelated topics for each of

three years of high school (Meiring et al., 1992; NCTM, 1989).

 

2Not coincidentally, the Connected Mathematics Project and the Core-Plus Mathematics Project
coordinated their efforts so that the Core-Plus (high school) curriculum would essentially pick up where the
Connected Math (middle school) curriculum leaves ofi‘.

Through what I knew of both these curricula, I held high hopes that therein might
lie some of the answers that I had sought as a novice high school math teacher trying to
help my students experience math in ways that didn’t simply leave them bored and
unengaged doing “naked number” problems for which numerous algorithms—understood
procedurally (at best)—must be memorized and performed in order to be successful.

Another significant part of my graduate studies at Michigan State was my work
on the Mathematics and Teaching through Hypermedia (M.A.T.H.) Project with Deborah
Ball and Maggie Lampert from 1993-1995. My work involved videotaping classes and
workshops taught by Ball and Lampert, organizing and preparing data in electronic
format for entry into an interactive database, as well as using their hypermedia materials
(Lampert & Ball, 1998) to teach several elementary math methods courses.

From these experiences, I came to think in a whole new way about what it means
for students to understand mathematics. I saw how fragile students’ seemingly “correct”
understandings can be. I witnessed firsthand the fact that, although students may give a
“right” answer, this does not necessarily indicate that their understandings are correct
(Erlwanger, 1975); I also came to appreciate the fact that, although students may give a
“wrong” answer, it does not necessarily indicate a complete lack of understanding or
mathematical insight (Ball, 1997; Duckworth, 1996; Heaton, 2000; Rosenthal, 1999).
Indeed, I grew more and more intrigued and interested in those students who did not
always have the right answers, those who seemed to struggle with their mathematical
understandings, those who were really trying to make sense of the concepts for
themselves (rather than just memorizing a set of steps to follow blindly). It was these

students whose mathematical insights had to be uncovered by eliciting their thinking

about problem-solving tasks so as to help them pursue the promising leads that, ofien,
they themselves didn’t even know they had stumbled onto (Duckworth, 1996;
Labinowicz, 1987; NCTM, 1991).

As I made my way through my doctoral program at Michigan State, I knew that I
wanted to make the Core-Plus curriculum the focus of my dissertation because, as a high
school mathematics curriculum, it was one that I knew truly made an effort to facilitate
students’ learning of mathematics in ways that also deepened their relational, rather than
just their instrumental, understandings of mathematics (Skemp, 1978). HOwever, the
curriculum was very new (first published for nationwide release in 1998), and I had
barely had the opportunity to look it over, let alone see how it played out in the
classroom. In the summer of 1997, I was able to attend a week-long conference for
Michigan math education leaders who were working on the effort to help implement the
Core-Plus curriculum throughout the state, which gave me my first actual glimpse at
some of the actual investigations and lessons. At the time, I figured that, for my
dissertation, I would undertake some kind of comparative (theoretical) curricular
analysis. Little did I know that just a year later I would have the opportunity to actually

try my own hand at teaching the curriculum.

Back to Teaching

During the 1998-99 school year, I found myself back in the high school math
classroom at a laboratory school on the Ball State University campus. Because of the
academic freedom of the laboratory-school environment, teachers were allowed to

exercise the prerogative to supplement or replace the selected textbooks for their courses.

I knew that my Algebra One class was considered a remedial section—a collection of
students who were placed in the course as ninth-graders (a year behind the majority of the
students at the school) because each had struggled with math in their previous school
years. They had been deemed not-yet—ready for algebra in eighth grade, or they had not
yet passed Algebra One in previous attempts. Indeed, one third of the students in the
class would be taking it for (at least) the second time. Eleven of 18 students in the class
were at risk of failing, or seriously struggling in, the course because of at least one (and
in most cases, several) factors such as absenteeism, repeated suspensions from school,
previous academic failure, learning disabilities, and so on.

The standard textbook for the course (Schultz, Hollowell, & Ellis, 1997), which
was on the state-approved list, struck me as an updated version (i.e., lots of color pictures
related to teen life in the late 19905) of the text I had used nearly a decade earlier in my
Pre-Algebra classes in Kansas (Willcutt, Fraze, & Gardella, 1984). I wondered: If these
students had not previously been successful with this kind of approach to learning
mathematics, then why would repeating the same curriculum be of help? So I decided to
use the Core-Plus curriculum in this class. The second and third units of the first-year
course (Coxford et al., 1998b) are primarily focused on algebra concepts; and the first
unit lays the groundwork for students to be able to do the data analysis (using graphing
calculators to interpret various representations of data in tabular and graphical forms) that
was essential to Units 2 and 3. So we set out to work on the first three units.

Although Course 1 in its entirety includes eight units of study covering algebra
and functions, statistics and probability, geometry and trigonometry, as well as discrete

math, my students and I stuck to the algebra-focused units—not only because the students

were enrolled in an algebra course, but also because, in keeping a pace that afforded
students time to master the concepts, we never got farther than Unit 3 by the end of the
school year. And, although it seemed unsatisfactory that the students had covered just
three of the four units of Part A of Core-Plus Course 1 for the school year, I witnessed
some truly satisfying results. I observed these “remedial” students’ success in making
sense of algebraic concepts (Skemp’s relational understanding), discussing them with a
mathematical sophistication that I had not seen even from more advanced students in

traditionally taught second-year algebra courses.3

My Consequent Research Interests

From this experience grew my research interests. Initially, this study was
designed to help answer the question, What kinds of understandings of algebra and
functions do previously unsuccessful students possess after studying Units 1, 2, and 3 of
Core-Plus Course 1? Understandings of algebra and functions was the figure being

studied; previously unsuccessful students was the ground:

 

3The entirety of Core-Plus Course 1 contains eight units—four units in Part A and four units in Part B.
While three units of Part A amount to just 40% of the entire Course 1 curriculum, the curriculum
developers have acknowledged that there will be some students who need two school years to complete
both parts of the Course 1 curriculum (Coxford et al., l998d, p. 4). Assuming that this pace was appropriate
for my Algebra One students, the three units of Part A that they completed amount to 80% of the Course
1A curriculum, which we supplemented with six weeks of traditional algebra skills work from the school’s
pre-selected text for Algebra One (Schultz et al., 1997).

 

 

UNDERSTANDINGS
OF ALGEBRA &
FUNCTIONS

 

 

 

Previously unsuccessful students

 

Figure 1.1
First iteration of research focus

As described in Chapter 4, my data collection began with a survey given to a large
group of students as a means of selecting a small number of study participants. The
survey had been carefully designed to help discern which students were previously
unsuccessful in school mathematics (according to criteria described in Chapter 3) based
on their entering attitudes toward the subject and its study. As soon as interviewing
commenced with the selected study participants, I began to inquire about their previous
non-successes in math. Indeed, every participant had something in her personal math
history that was qualitatively unsuccessful. The variety among these previous non-
successes was quite interesting and insightful, and my research focus shifted slightly. I
became as interested in the participants’ previous non-successes in school mathematics as

I was in their understandings of algebra and functions:

 
     
   
 

 
 
    
 

  

Previous non-
successes in school
mathematics

  

Understandings of
algebra & functions

Figure 1.2
Second iteration of research focus

As a part of my interview protocol, I had included questions about students’
learning experiences in Core-Plus Course 1. These questions were meant to help detect
whether participants’ entering attitudes toward school mathematics were changing over
time. Each time I asked these questions of a participant, I took more and more interest in
her learning experiences in Core-Plus Course 1. As I collected these data, the
observations 1 was making about students’ learning experiences seemed, in some ways, to
be the bridge that I used to connect students’ previous non-successes in school
mathematics with what students were coming to understand about algebra and functions
during their studies in Core-Plus Course 1. In other words, students’ experiences of the
Core-Plus Course 1 curriculum became the vehicle through which I was beginning to
make sense of their understandings of algebra and functions (Erickson & Shultz, 1992).
Yet, at that time, students’ learning experiences played a secondary role to the two main

components of my inquiry.

 
  

 
 
    
 

     
   

Previous non-
successes in school
mathematics

 

  

Understandings of

Core-Plus Course 1
learning experiences algebra & functions

 

Figure 1.3
Third iteration of research focus
During the final phase of my research, in the course of trying to make sense of all
my data, I came to fully appreciate the importance of students’ learning experiences in
Core-Plus Course 1. It has shown itself to be a third focal point, meriting equal attention

with the other two:

 
  

 
 
     

Understanding
algebra & functions

Previous non-successes Core-Plus Course 1
in school mathematics learning experiences

 

Figure 1.4
Final iteration of research focus
The three prongs of this inquiry are interrelated. Each informs the other two and cannot
be fully explained or understood without them.

In this final iteration of my research focus, there has also been a subtle but
important change in the framing of the exploration of students’ understandings of algebra
and functions. In the diagram above, note that the focus formerly worded
“Understandings of algebra and functions,” is now worded “Understanding algebra and
functions.” “Understandings” as a noun has been replaced by “understanding” as a verb.
This is intentional. It reflects the ongoing nature of coming-to-understand things.
Understandings, as any constructivist teacher of mathematics knows, are fragile and ever
changing—especially in the beginning stages of learning new ideas, skills, and concepts.

Understanding is a learning process. This study is but one small snapshot of that process.

My Questions
Stated as questions, the three prongs of inquiry that compose this study are:
I What is the nature of these students’ previous non-successes in school

mathematics?

10

- What are these students’ experiences of the Core-Plus curriculum?
. What do these students understand about algebra and functions (at three different

moments in time)?

11

CHAPTER 2

OVERVIEW OF MATHEMATICS-EDUCATION AND ALGEBRA REFORM

Everybody counts

The last two decades of the 20th Century have proven to be a significant period of
reform in school mathematics in the United States. This movement was driven by a
number of factors including revolutionary changes in the work world brought about by
the advent of technology (Johnston & Packers, 1987; Secretary’s Commission on
Achieving Necessary Skills, 1990) and reports about the miserable state of mathematical
knowledge of schoolchildren in the US. compared to other nations worldwide (National
Commission for Excellence in Education, 1983). Led by the National Council for the
Teachers of Mathematics (NCTM), math educators across the country began to
reconsider what mathematics is important for schoolchildren to learn (N CTM, 1980) as
well as what teaching approaches are most effective in helping students achieve both
computational and conceptual understandings of the mathematics they study in school.

One of the fundamental issues underlying this reform movement has been the
concern for those students who, for a variety of reasons, have been historically
disadvantaged in or, worse yet, filtered out of mathematics as a field of study, as a career
option, as a subject in school in which they can succeed—females, students of color,
students from lower socioeconomic backgrounds, special-needs students. “Opportunity
for all” was a defining goal of the original Standards document, which declared,

The social injustices of past schooling practices can no longer be tolerated.

Current statistics indicate that those who study advanced mathematics are most

often white males. Women and most minorities study less mathematics and are
seriously underrepresented in careers using science and technology. Creating a

12

just society in which women and various ethnic groups enjoy equal opportunities

and equitable treatment is no longer an issue. Mathematics has become a critical

filter for employment and firll participation in our society. We cannot afford to
have the majority of our population mathematically illiterate: Equity has become

an economic filter. (NCTM, 1989, p. 4)

At the same time that NCTM published its original Standards document, the
National Research Council released its landmark publication Everybody Counts: A
Report to the Nation on the Future of Mathematics Education (1989). The key theme for
which this document is best known is its declaration, “Mathematics must become a pump
rather than a filter in the pipeline of American education” (p. 7). That is to say that,
rather than weeding out and thus preventing all but the most gifted. or advantaged
students from pursuing mathematics as a field of study and as a career choice, the way in
which mathematics is taught and learned in our society must become such that it enables
and empowers all students to succeed as mathematically literate participants in society
(NCTM, 1989; NRC, 1989).

While Everybody Counts served as an outline of the philosophical underpinnings
of the burgeoning mathematics reform movement, the first NCTM Standards document
provided the practical recommendations for changes needed in the content and emphasis
of school mathematics. These recommendations called for decreased the attention paid to
complex and/or tedious paper-and-pencil computations; rote practice and memorization
of rules, formulas, and facts without understanding; isolated treatment of arithmetic
procedures and mathematical topics; and developing skills out of context.
Recommendations were also included to increase emphasis on developing thinking,

reasoning, and problem-solving strategies; understanding of the meanings of numbers,

operations, and the interconnectedness of math topics; recognizing patterns and

13

relationships; being able to communicate mathematically and work collaboratively with
peers; and using a variety of methods, including technology and mathematical models, to
solve problems (1989, pp. 20-21, 70-73).

The second of NCTM’s Standards documents (1991)l summarized the changes in
instructional practice needed to bring about these results:

Woven into the fabric Of the Professional Standards for Teaching Mathematics
are five major shifts in the environment of mathematics classrooms that are
needed to move from current practice to mathematics teaching for the
empowerment of students. We need to shift —

0 toward classrooms as mathematical communities—away from classrooms
as simply a collection of individuals;

0 toward logic and mathematical evidence as verification—away from the
teacher as sole authority for right answers;

0 toward mathematical reasoning—away from merely memorizing
procedures;

0 toward conjecturing, inventing, and problem solving—away from an
emphasis on mechanistic answer-finding;

0 toward connecting mathematics, its ideas, and its applications—away from
treating mathematics as a body of isolated concepts and procedures. (p. 3)

Algebra for all

A major focus of reform at the secondary level has been directed at the teaching
and learning of algebra. For decades, the traditional ninth-grade algebra course has been
the critical filter for high school mathematics students (Chazan, 1996; Moses, 1994;
Moses & Cobb, 2001; Usiskin, 1993, 1995). The prototypical algebra course—in which
the skills practice focuses heavily on symbolic manipulation and the use of algorithms to
solve problems out of context, equations and functions must be graphed by hand (using
only paper and pencil), and classroom instruction is exclusively characterized by an

observational-leaming model whereby the teacher lectures and demonstrates problem-

 

1Two additional volumes (NCTM, 1995, 2000) round out the set of Standards documents.

14

solving routines that individual students imitate—is a realistic description of the high

school mathematics experience most adults in the United States have known.

The Algebra Working Group, convened by the NCTM in the mid-19905 to rethink

the algebra curriculum, summarized the need for algebra reform:

The algebra experience for all students in grades K-12 must be greatly improved.
All students need the tools and mathematical thinking fostered by the study of
algebra to function effectively in a dynamic and challenging world. Currently
many students are being disenfranchised by lack of access to algebra as well as
an impoverished curriculum. Algebra needs to change. (Burrill et al., 1994, p. 3)

Among the primary tenets of its work, the Algebra Working Group asserts that

“[t]echnology changes the way algebra should be taught, learned, and used” and that

“[t]he world presents problems and challenges that demand a different kind of algebraic

knowledge” (p. 4). The recommendations made by the Algebra Working Group built

directly upon the reform vision of algebra put forth by the NCTM in 1989 (and further

refined in 2000):

Instructional programs from prekindergarten through grade 12 should enable all
students to —

Understand patterns, relations, and firnctions

Represent and analyze mathematical situations and structures using
algebraic symbols

Use mathematical models to represent and understand quantitative
relationships

Analyze change in various contexts. (NCTM, 2000, p. 296)

Specific recommendations for changes in the content and emphasis of the study of

algebra and functions at the high school level appeared in the original NCTM Standards

document:

15

 

Topics to receive increased attention

Topics to receive decreased attention

 

ALGEBRA

The use of real-world problems to motivate
and apply theory

The use of computer utilities to develop
conceptual understanding

Computer-based methods such as
successive approximations and graphing
utilities for solving equations and
inequalities

The structure of number systems

Matrices and their applications

ALGEBRA

Word problems by type, such as coin,
digit, and work

The simplification of radical expressions
The use of factoring to solve equations
and to simplify rational expressions
Operations with rational expressions
Paper-and-pencil graphing of equations
by point plotting

Logarithm calculations using tables and
interpolation

The solution of systems of equations
using determinants

Conic sections

 

 

FUNCTIONS

Integration across topics at all grade levels
The connections among a problem
situation, its model as a function in
symbolic form, and the graph of that
function

Function equations expressed in
standardized form as checks on the
reasonableness of graphs produced by
graphing utilities

Functions that are considered as models of
real-world problems

 

FUNCTIONS

Paper-and-pencil evaluation

The graphing of functions by hand using
tables of values

Formulas given as models of real-world
problems

The expression of function equations in
standardized form in order to graph
them

Treatment as a separate course

 

 

(1989, pp. 126-127)

Reform-based secondary mathematics curriculum and the Core-Plus Mathematics
Project

In addition to the recommended curricular changes in content and emphasis, the
1989 Standards document also introduced a new paradigm for the structure of high
school mathematics: the core curriculum (NCTM, 1989, pp. 123-136). The “core topics”
include not only the traditional topics of algebra, geometry, trigonometry, and functions,
but also statistics, probability, and discrete mathematics. It is emphasized that “the core
curriculum is intended to provide a common body of mathematics ideas accessible to all

students” (p. 123). Moreover, the idea of a core curriculum was proposed with the

16

“expectation that mathematical ideas will grow and deepen as students progress through
the curriculum” (p. 124).

While the 1989 Standards document offers guidelines for the content of a core
curriculum in the form of fourteen curricular standards for Grades 9-12 (pp. 137-186), it
does not provide many practical suggestions. However, in 1992, as part of its Addenda
Series for Grades 9-12 (Meiring, Rubenstein, Schultz, de Lange, & Chambers, 1992),
NCTM did provide “several possible curriculum models for organizing the mathematics
content recommended in the Curriculum and Evaluation Standards” (p. vi). Still, only
nine sample lessons for each of two curriculum models are provided in the Core
Curriculum Addenda booklet. Thus, it has been left to curriculum developers to design
instructional materials and texts that exemplify the core curriculum envisioned by the
NCTM:

In 1990, the National Science Foundation (NSF) responded to the call for new

materials by funding curriculum development projects at the elementary school,

middle school, and secondary school levels. At the secondary level, five projects
were funded. . ..All five secondary school mathematics projects are currently

available as multiyear textbook series. (Martin et al., 2001, pp. 540-541)

The Core-Plus Mathematics Project (CPMP), which developed the textbook series
Contemporary Mathematics in Context, is one of the five secondary school projects that
the NSF funded. The Core-Plus curriculum is a four-year integrated mathematics
program (Coxford et al., 1998d, p. 1). The first three courses are intended to be studied
by all students, the fourth course by college-intending students. Each of the first three
courses comprises seven units of study followed by a “culminating capstone experience”

that

...is a thematic, two-week project-oriented activity that enables students to pull

l7

together and apply the important modeling concepts and methods developed in
the entire course. (Coxford et al., 1998d, p. 4)

Throughout each of the seven units of Courses 1-3, four content strands are
interwoven, which were designed to address the curriculum standards for Grades 9-12
(NCTM, 1989). Some or all of the content strands—algebra and functions, geometry and
trigonometry, statistics and probability, and discrete math—may be incorporated into any
given unit with various degrees of emphasis (Coxford et al., 1998d).

Also in keeping with the recommendations of the NCTM Standards (1989), the
instructional approach of the Core-Plus program differs from the traditional model that
has been known to characterize high school math classes for decades prior to the reforms
that have taken place in the 19808 and 19903:

0 Students focus on real-life problem situations from which new mathematical skills
emerge——in contrast with the traditional skills focus on symbolic manipulation and
algorithms to solve problems that lack any context;

0 Students employ graphing calculators to examine multiple representations of data and
functions—in contrast with the traditional approach which prioritizes solving and
graphing equations using only paper and pencil;

0 Students tackle mathematical investigations in small groups followed by a teacher-
guided whole-group discussion of the mathematical underpinnings—in contrast with
the traditional approach that emphasizes an observational-learning instructional
model where the teacher lectures and demonstrates algorithmic problem-solving

routines that individuals imitate.

l8

The “math wars "

Reformed math curricula have not been embraced with open arms by all,
however. The Core-Plus developers themselves are well aware of the backlash:

The spirited debates about the reform of school and undergraduate mathematics

have led some proponents and opponents of change to indulge in such angry

rhetoric that the controversy has come to be referred to as the “math wars.”

(Schoen, Fey, Hirsch, & Coxford, 1999, p. 445)

These debates date as far back as 1995, when a number of concerned parents
organized to oppose the math curriculum being used in California schools. The middle
school math curriculum at issue was adopted by the Palo Alto school district because it
conformed to the state’s math frameworks, adopted in 1992, which were based upon the
1989 NCTM Standards recommendations for reforming school math instruction. Shortly
thereafter, a rival group of parents who favored the new math curriculum rose up to show
their support (Sommerfeld, 1996; West, 1995a, 1995b). Not quite a year later, “similar
battles [were] being fought in other states, from Iowa to Texas” (Sommerfeld, 1996, p. 1).

[J ]ust as the new curricula, teaching methods, and assessment strategies are

beginning to be tested in schools and universities across the country and are

beginning to show promise of reaching the objectives of reform, critics have
challenged the content goals, the pedagogical principles, and the assessment
practices that are at the heart of the reform agenda. What seemed to be an
overwhelming national consensus on directions for change in mathematics
education is now facing passionate resistance from some dissenting
mathematicians, teachers, and other citizens. Wide dissemination of the
criticisms—through reports in the media, through Internet mailings, and through
debates in the meetings and journals of mathematics professional societies—has

shaken public confidence in the reform process. (Schoen et al., 1999, pp. 444-

445)

The critics—including everyone from parents and teachers to national

politicians—charge that the reform curricula have done away with the learning of “basic

skills.”2 They also oppose the student-centered nature of the reform approach, and
deplore the deviation from standard-looking textbook fare of many reform curricula
(Loveless, 1997).

The battle rages on, and opponents gained new fervor when the federal
Department of Education released a list in late 1999 naming ten reform curricula as
“exemplary” or “promising” (Viadero, 1999a, 1999b). Even national legislators were
drawn into the dispute after “nearly 200 mathematicians, physicists, and other scholars
took out a full-page advertisement in The Washington Post to air their opposition” to the
list of endorsed math curricula (Klein, 2000; Viadero, 2000).

The Core-Plus curriculum has not been exempt from the “math wars”
controversy. For instance, in one of the schools that piloted Core-Plus, there were widely
contrasting opinions as to whether high school students who studied the Core-Plus
curriculum were adequately prepared for their college mathematics courses (Clayton,
2000a)

The criticism leveled at Core-Plus increased when it appeared on the Department
of Education’s list of “exemplary” curricula:

The Department of Education’s expert panel, assembled to find the best math

programs in the United States, had a key problem, critics say. It relied heavily on

studies of student achievement that were authored or co-authored by the directors
of the programs themselves—or by people with close institutional or other ties to

the program. (Clayton, 2000b, p. 15)

The chief complaint of critics was that results of field tests of the curricula—

Core-Plus and Connected Math (Lappan, Fey, Fitzgerald, Friel, & Phillips, 1998), in

particular—had not been subject to independent evaluation such as that required for

 

2Indeed, the 2000 NCTM Principles and Standards was written in large part for the purpose of a détente in
the “math wars” (Hoff, 2000).

20

publication in peer-reviewed journals (Clayton, 2000b).

Further research needed

Despite the fact that the lack of independent evaluation does not render invalid the
studies that were done by program-affiliated researchers (Clayton, 2000b), the Core-Plus
curriculum developers have acknowledged the need for further research on the
effectiveness of reform curricula in general (Huntley, Rasmussen, Villarubi, Sangtong, &
Fey, 2000). They continue to carry out their own studies on the effects of Core-Plus
(Schoen, Cebulla, & Winsor, 2001; Schoen, Finn, Griffin, & Fi, 2001; Schoen, Hirsch, &
Ziebarth, 1998; Schoen & Pritchett, 1998; Ziebarth, Slezak, Lagrange, & Kleinfelter,
1997) while inviting others to join them in the process of evaluating the comparative
effects of a reformed high school math curriculum to that of a traditional curriculum:

The broad purpose of this study was to test the vision of reform proposals in
recent advisory documents like the NCTM Standards by comparing effects of a
curriculum designed to implement the Standards to those of more conventional
curricula. We collected and analyzed extensive data on student learning of algebra
from both kinds of curricula and found considerable support for main themes of
the reform. However, no single study will provide complete or conclusive
evidence.

Our study suggests some important patterns of consequences from
curricular, instructional, and assessment practices in high school mathematics.
Those patterns suggest areas in which both reform and traditional curricula need
to be improved if they are to reach widely agreed-upon goals. But they also leave
open the fundamental questions about what understanding and skill in algebra is
most important for students to acquire from their school mathematics experience.
Furthermore, they suggest some aspects of both reform and traditional curricula
that need to be studied in more depth with methods other than those used in this
study. (Huntley et al., 2000, pp. 359-360)

21

Occupying a unique spot in the existing literature

In my search to find other studies on the Core-Plus curriculum that have been
shared with and reviewed by the public to date, I have found almost one-and-a-half
dozen3—two studies published in peer-reviewed journals (Huntley et al., 2000; Wilson &
Lloyd, 2000), eight Ph.D. dissertations (Hetherington, 2000; Kahan, 1999; Kett, 1997;
Latterell, 2000; Lloyd, 1996; Truitt, 1998; Tyson, 1996; Walker, 1999), six papers
presented at major conferences (Schoen, Cebulla, & Winsor, 2001; Schoen, Finn, et al.,
2001; Schoen et al., 1998; Schoen & Pritchett, 1998; Van Zoest & Bohl, 2000; Ziebarth
et al., 1997), and one master’s thesis (Kohmetscher, 2001). Ten of the 17 study reports
have at least one author who is directly affiliated with the CPMP, either as a member of
the curriculum project team or as an instructor who helped pilot the curriculum.4 Of the
seven reports whose authors are not directly affiliated with the CPMP (Hetherington,
2000; Kahan, 1999; Kohmetscher, 2001; Lloyd, 1996; Tyson, 1996; Van Zoest & Bohl,
2000; Wilson & Lloyd, 2000), all but Hetherington and Kohmetscher could certainly be
considered indirectly linked to the project.5

Of these 17 Core-Plus studies, 8 focus the lens of research on students, 8 focus on

teachers, and 1 (the master’s thesis) focuses on the parents’ role in the Core-Plus

 

3This number represents publicly reviewed presentations of distinct studies. A “publicly reviewed”
presentation includes an article published in a peer-reviewed journal, a paper presented at a major
conference, or a dissertation/thesis. Multiple presentation modes of the same study are not included.

’As mentioned in the previous section, one of the most pointed criticisms that was leveled at the Core-Plus
curriculum when it received its “exemplary” rating fi'om the federal Department of Education in Fall 2000
was precisely this: that evaluations of the program’s effectiveness were co-authored by those directly or
indirectly affiliated with the CPMP (Clayton, 2000b).

5 Melvin (Skip) Wilson was a longstanding colleague of the late Art Coxford at the University of Michigan,
who served as one of the co-directors of the CPMP, and Wilson served as the major advisor for Lloyd’s
doctoral dissertation. Laura Van Zoest is a colleague of co-director Christian Hirsch at Western Michigan
University and served as the major advisor for Rebecca Walker’s doctoral dissertation (a CPMP team
member mentioned above). Tyson completed her doctorate at the University of Iowa, where co-director
Henry Schoen is on the faculty, and Kahan completed his doctorate at the University of Maryland where
co-director Jim Fey is based.

22

curriculum. The eight studies that address students’ issues (Huntley et al., 2000; Kahan,
1999; Latterell, 2000; Schoen, Cebulla, & Winsor, 2001; Schoen et al., 1998; Schoen &
Pritchett, 1998; Tyson, 1996; Walker, 1999) are all quantitative in nature, each involving
several hundred students as study subjects (taking tests, completing surveys, or providing
college placement exam scores). Two of these student-focused studies examine students’
preparation for, or transition to, college mathematics (Schoen, Cebulla, & Winsor, 2001;
Walker, 1999). Walker (1999) also examines students’ (epistemological) conceptions of
mathematics; while Schoen and Pritchett (1998) examine students’ (epistemological)
perceptions of and (affective) attitudes towards mathematics. Tyson (1996) explores
gender differences on performance tests. Schoen et a1. examine students’ quantitative
reasoning, Latterell (2000) examines students’ general problem-solving ability, Kahan
(1999) examines students’ competence with proof, and Huntley et a1. (2000) are unique
in their focus on students’ “understanding, skill, and problem-solving ability in algebra”
(p. 328, emphasis added).

Given this map of the current Core-Plus research literature, my research occupies
a unique spot. It joins the ranks of the eight studies that focus on students’ issues, yet I
am the only student-focused researcher who is neither directly nor indirectly affiliated
with the CPMP. Mine is the only study to focus exclusively on those students who have
been previously unsuccessful in school mathematics.6 Moreover, to my knowledge, I am
the first to examine students’ learning using qualitative methods. This has allowed me to
explore issues that other researchers have undertaken—students’ understandings of

algebra and functions; and students’ attitudes towards mathematics (which is part of the

 

°A complete explication of what is meant by “previously unsuccessful” students is included in Chapter 3.

23

exploration of students’ previous non-successes in school mathematics)—yet paint them
in different hues by focusing on just six students.
Perhaps most important, however, is the new perspective that I bring. My study is
novel in its exploration of students’ experiences of the curriculum.
[V]irtually no research has been done that places student experience at the center
of attention...If the student is visible at all in a research study he is usually viewed
from the perspective of adult educators’ interests and ways of seeing, that is, as
failing, succeeding, motivated, mastering, unmotivated, responding, or having a
misconception. Rarely is the perspective of the student herself explored.
Classroom research typically does not ask what the student is up to, nor does it
take a critical stance toward its own categories and assumptions so as to question
whether “failing” or “mastering” or being “unmotivated” or “misconceiving”
adequately captures what the student might be about in daily classroom
encounters with curriculum. (Erickson & Shultz, 1992, pp. 467-468)
Holt et a1. (2001) have taken a first step in presenting first-hand accounts of students’
experiences with a reform high school mathematics curriculum. My approach is to
present third-person reports of students’ experiences with such a curriculum,
incorporating the added information of students’ past experiences in school mathematics,
while focusing on a particular strand of content leaming—algebra and functions. While I
do not claim to present a study that examines both the social participation and subject
matter task structures in depth (Erickson & Shultz, 1992), I believe I do take a first step
toward offering the kind of investigation that Erickson and Shultz advocate:
Presently it is not at all clear how social relations with the teacher and with fellow
students become part of the math problem the student encounters, nor do we know
how the student’s “archaeology” of past experience with mathematics in both its

social and intellectual aspects enters into the experience of the academic task
structure of the moment at hand. (1992, p. 475)

24

CHAPTER 3

CONCEPTUAL FRAMEWORK

Before venturing any further to describe this study, it is important to offer a
synopsis of my own ideological stance on the issues that play a major role in shaping the
direction(s) of my research in order for the reader to better understand the terrain of

mathematics education reform that I plan to explore.

Issues related to math and algebra reform

Perhaps the most significant of these issues is the belief that mathematics
instruction (and algebra instruction, in particular) must effectively reach all students
(Chazan, 1996; Hirsch & Coxford, 1997; NCTM, 1989, 2000; Secada, 1992; Usiskin,
1993, 1995). “All students” includes those typically and historically filtered out of their
study of mathematics —— those students who are tracked into a lower level of mathematics
instruction in school, including (especially) girls, students of color, and, very often,
students who do not enjoy the socioeconomic and educational advantages that are
common to middle- and upper-class students (National Research Council, 1989).

To “effectively reach” all students, many believe and have argued that,
mathematics instruction (and algebra instruction, in particular) must take on new forms,
different from the traditional approach (Chazan, 2000; Moses & Cobb, 2001; NCTM,
1991). Chazan argues that non-traditional approaches to first-year algebra instruction are
imperative because the traditional curriculum drives out lower-track students and

deprives them of conceptual understandings. Moses makes a similar case: Algebra is a

25

gatekeeping course whereby students, who do not succeed, are denied access to equal
participation in a technologically oriented society (which he considers a civil right as
significant as the right to vote); moreover, algebra is accessible to all students only when
it can be understood conceptually, learned experientially, explored within contexts that
are meaningful to students.

Another key to effectively reaching all students lies in the type of understanding
that has been chosen as the goal for mathematics education (Skemp, 1978). Skemp
makes a strong argument in favor of relational understanding1 as a better goal for
mathematics education than instrumental understanding. It is students’ relational
understanding (knowing what to do and why) that I am investigating in this study rather

than their instrumental understanding (merely knowing what to do).

Issues pertaining to “previously unsuccessful ” students

The students whom I have chosen to make the focus of my study are a subset of
those whom I have dubbed “previously unsuccessful.” This is a term I have selected to
describe students who have been alienated in their earlier study of mathematics. A
student’s feelings of alienation might be linked to her/his gender, ethnicity,
socioeconomic status, language background, special needs, learning style, work ethic, or
a poor experience in a previous math class; the reason behind students’ alienation was not
of particular importance to me, however. Instead, it was those characteristics that
unsuccessful students commonly exhibit that held the key to identifying those who would

make the best candidates for study participants.

 

lBuxton (1978) calls this “insightful” understanding, and Hiebert et al. in Hiebert (1986) refer to this as
“conceptual” understanding.

26

House (1988) describes seven components of success for students in math and
science that were particularly useful in helping me pinpoint identifying characteristics of
(previously) unsuccessful students. Of seven characteristics discussed by House, I chose
four—two of which were broken into subcategories—that I felt could be detected through
students’ self-reports on a written questionnaire. The other three characteristics are ones
that I felt students would not be able to report themselves, but could only be discerned
through my first-hand observation of and personal interviews with the students.
Qh_ar;a£teristics to be determined from students’ self-reports

SELF-CONFIDENCE: The student believes she can learn. She doesn’t feel
helpless, and she feels math is for her.

Public. This is the confidence a student possesses about her own skills

when those skills can be viewed by others.

Personal: This is a student’s private knowledge of how well she can do.

ENTHUSIASM: The student possesses a sense of ownership, curiosity, drive
to learn. She possesses a respect for math and technology; a willingness to
think creatively, to take risks, to subject her ideas to the scrutiny of others; and
an excitement about learning math.

SIGNIFICANCE: The significance of math is not lost on the student. She
sees applications of math in her everyday life. She views the world through
math-colored glasses and possesses an awareness of the interrelatedness of

math, science, and technology.

27

STICK-TO-IT-IVENESS:

Long-range: The student plans to take more math (and math-related)

courses and those that are more challenging than what is required.

Short-range: The student possesses perseverance. She stays on task and

isn’t afraid to grapple with important ideas or hard problems. The student

doesn’t necessarily need answers quickly, doesn’t give up easily, and is

not easily distracted. She is willing to think and struggle, to trust her own

judgments, and to risk being wrong.
Because House describes components of success for students in mathematics and science,
the lack of the above-named characteristics were taken as indicators of non-success.
C_hagicteristics to be determined from personal interviews and observations

Although House (1988) discusses another three characteristics of success
(understanding, competence, and communication), I have compiled a set of my own
indicators of success — characteristics to be determined from my personal interviews with
and classroom observations of the study participants. These indicators were born early in
my data analysis and were used throughout the data analysis process. While I do not take
a formal inventory of these characteristics for each participant, the presence or absence of
these indicators was helpful in discerning each participant’s overall success throughout
Core-Plus Course 1A.2

These indicators of success fall into two general categories, which are related to

two of my research questions.

 

2Here, and throughout the dissertation, I use “1A” when referring specifically to the first half of the
curriculum only (as in “My study is about Course 1A”); otherwise, when referring in general to the first
course in the Core-Plus curriculum (as in “Joy was enrolled in Course 1 as a ninth-grader”), I use just “1.”

28

ATTITUDES TOWARDS MATHEMATICS: These are related to participants’
learning experiences in Core-Plus Course 1. The component parts of this category
that proved to be of particular interest include
Satisfaction: The student feels positively about her mathematics learning
experience. She derives a certain amount of fulfillment and/or enjoyment from
her math studies.
Change in attitude: The student’s dispositions towards math, over time, become
more like those expected of successful students.
MATHEMATICS ACHIEVEMENT: This is related to participants’ understanding
algebra and functions throughout Core-Plus Course 1A. The component parts of this
category that proved to be of particular interest include
Text interpretation: The student understands the nature of a posed problem -
what it is asking the problem-solver to find out and what is involved in solving it.
Eggplanation of thinking: The student is able to verbally articulate her
mathematical thinking about a problem. She can explain her problem-solving
process to another person in such a way that he can understand her strategies and
insights.
Mathematical abstraction: The student is able to move beyond the problem
context to comprehend the mathematical principle(s) involved in the problem.
Retention: The student retains (or is able to recall without too much difficulty)
and is able to utilize mathematical concepts and skills learned previously.
Invention: The student makes successfiil use of alternate means of problem

solution.

29

The characteristics described above are indicators of success for students in mathematics;
therefore, the lack of these characteristics are taken as indicators of non-success.

Finally, it is important to note that, according to these criteria, “previously
unsuccessful” does not necessarily refer to the grades 3 student has received in her
mathematics classes, nor does it necessarily refer to a student’s conceptual grasp of the
subject matter. A student may receive good grades, yet still feel estranged from her math
studies if she is bored with or uninterested in the subject, if she sees no relevance of her
math studies to her everyday life, or if she has come to believe that mathematics is simply
not for her. Likewise, a student may also possess some propensity for mathematics
learning and demonstrate keen insight about mathematical concepts, yet she may
nonetheless feel disaffected by her math studies if she believes her math skills are inferior
to others’, if she sees no practical reason for having to pass the required math classes in
order to graduate high school, or if she sees no value in pursuing a difficult problem to its

resolution.

A map of the algebra terrain

In order to answer the question of what these students understand about algebra
and functions, it is also necessary to map out the terrain that comprises algebra, to stake
out the territory that is the mathematical focus of this research. The map included here
represents a high school math teacher’s view of algebra, drawn from the canon Of various
19808 and 19908 US. high school math curricula, from traditional to reformed, as well as
the NCT M Standards documents (Coxford et al., 1998b; Dolciani et al., 1984; Dolciani et

al., 1986; Larson, Kanold, & Stiff, 1997; Murdock, Kamischke, & Kamischke, 1998;

30

NCTM, 1989, 2000; Schultz et al., 1997).

Six fields of study within algebra are identified: multiple representations,
functions, levels of abstraction, number relationships, means of solution, and number
groups. No hierarchy is implied in the listing of these fields; indeed, they are inter-
connected and are typically studied as such, with various elements from each field
introduced and revisited at various times throughout the course of one’s studies of high
school algebra. It should be noted, however, that more traditional high school curricula
tend to introduce elements, and even some fields, in a more linear and less integrated
fashion than do the more reformed curricula.

A8 a means of constructing this map of the algebra terrain, I began by listing all
the topics I had taught as a high school math teacher of Algebra One, Algebra Two, and
Core-Plus Course 1A. I also considered what algebra topics are necessary for subsequent
mathematics courses (trigonometry, precalculus, calculus, statistics) as well as for other
“client courses” such as chemistry and physics. Finally, I also considered what sorts of
mathematical skills were included on college entrance exams (SAT and ACT) and on
college math placement tests. The resulting set of topics is the algebra map that you see
here, after being organized and arranged into the fields of study listed above.

This algebra map is offered with specific purposes in mind and is, admittedly,
limited in its utility. First of all, this map is presented as a means of broadly sketching
out the territory of high school algebra studies in the United States over the last two
decades. A macroscopic view is presented; as such, it is meant as an aid in orienting
yourself, while exploring the question of what previously unsuccessful students

understand about algebra and functions, in order to be able to locate the region in which a

31

student’s algebraic understandings lie.

Secondly, it is not to be inferred that a mapping of this type is meant to convey
the entire picture of what algebra learning is all about. A listing of topics cannot address
the sum of what students should know and be able to do from their study of high school
algebra (Schoenfeld, 1994). Certainly, there are concepts and skills, mathematical ways
of knowing and doing, as well as connections between topics, included in the study of
high school algebra, that are not (and could not be) portrayed here.

Thirdly, this algebra map is not intended to represent a vision for what school
algebra could or should be (a project that lies far beyond the scope of this dissertation!)
This map is not intended to reflect any particular vision of algebra that has been put forth
(e.g., Chazan, 1996, 2000; Edwards, Jr., 1990; Lacampagne, Blair, & Kaput, 1995;
NCTM, 1989; Usiskin, 1988, 1995); nor does it represent my own vision for what school
algebra could or should be. The map presented here is simply meant as a tool for

analysis, a means to an end, and not as an end in itself.

Issues not addressed in this study
There are several issues—related to gender, social class, and immigrant
students—that I have deliberately chosen not to address in this study. Because of who
the study participants turned out to be, any one (or all) of these issues could have been
addressed yet, in the interest of limiting the scope of my analysis, I have set these issues
aside and have focused solely on those issues that are raised in my research questions.
First, all participants were girls, and there are significant concerns about girls and

mathematics learning discussed in the reform literature (E.g., Clewell, Anderson &

32

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33

Thorpe, 1992; Fennema & Leder, 1990; Secada, 2000; Secada, Fennema & Adajian,
1995). However, much more investigation of these particular issues would have to have
been planned into the study and, as it was, such issues did not present themselves and are
therefore not included in the final analysis of the data.

Second, among the very limited data that I gathered about participants’ personal
backgrounds, there are indications that several of them were likely related to factors
having to do with social class, and there are significant concerns about the role of
“cultural confusion” (Thuele-Lubienski, 1996) concerning, in particular, students of
lower-SES standing in the reformed mathematics classroom. However, given the nature
of my inquiries, I did not have access to data related to participants’ socioeconomic
statuses. Thus, these issues do not play a role in my analysis of the data.

Lastly, one participant (Monica) was, in fact, a recent immigrant to the United
States, and there were several indications that issues pertinent to immigrant students
played a role in her school experience. However, exploring these issues was well beyond

the scope of this study. Therefore, these issues are not mentioned in my analysis of the

data.

34

CHAPTER 4

METHODS OF DATA COLLECTION AND ANALYSIS

Research Context
Selecting_a site for my research

I sought, as a site for my research, an Indiana school district that uses the Core-
Plus curriculum in its high school(s). Because of my interest in studying previously
unsuccessful students’ understandings of first-year algebra concepts, it was necessary to
find a district that did not track its lower-achieving students into course offerings other
than Core-Plus. Through the use of a massive on-line database maintained by the Indiana
Department of Education (2002), I was able to identify a single qualifying school district
in the state of Indiana whose district included six public high schools with an average
enrollment of just over 1400 students per school.

Another parameter for locating a suitable research site was a school where the
math teachers had “bought into” the Core-Plus curriculum and who had experience
teaching Core-Plus in the spirit in which it was intended by the curriculum developers.
This parameter served to narrow the choices of potential school sites to one or two. By
time my request to conduct research within the school district was approved by the
teachers’ union discussion committee, I was given permission to work at only one school,
which I will call Oliver High School. At Oliver, there were two teachers teaching the
math classes in which students studied the first three units of Core-Plus Course 1 during

the first half of the school ear.l Between these two teachers, there were ei ht class
Y g

 

lOliver High School observed a Block-4 schedule during the 2001-02 school year. Under the Block-4 plan,
students took four full-year courses during the fall semesters which ran from mid-August to late October

35

sections with a combined enrollment of 150 students.
The Community

The community where Oliver High School is located, which I will call River City,
is a pleasant mid-size midwestern city.2 River City has garnered several national awards
recognizing its desirable characteristics such as a cost of living that is 7% below the
national average, a property crime rate that is 20% less than the national average, and a
violent crime rate that is about half the national average (Find Your Spot, 1999). River
City’s major employers include three hospitals (employing 7,000), two automotive
divisions (employing a combined 5,500), the River City school corporation (which
employs 3,800), a national insurance company headquarters (employing 3,400), a
telephone company (employing 2,500), and an electric motor manufacturer (1,800
employees). The city is home to four minor league sports teams, twelve institutions of
higher education (the largest of which is a public university enrolling nearly 14,000
students), several museums, at least three civic theaters, and a children’s zoo, as well as

2,200 acres of public parks and playgrounds.

 

and from late October to early January. Similarly, four full-year courses were taken during the spring
semesters which ran from early January to mid-March and from mid-March to the end of May. It was
arranged, by the Oliver High School Math Department, for the Core-Plus Course 1 curriculum to be taught
for three of the four semesters that comprise a nine-month school year: Students study three units of the
Core-Plus Course 1 curriculum for half the school year, attending class every other day (which equals one
semester), combined with the study of the other five units which are studied for half the school year,
attending class every day (which equals two semesters). Half of the students taking Core-Plus Course 1 in a
given school year attend math class every day in the fall, studying Units 1-5; these students attend math
class every other day in the spring, studying Units 6-8. The other half of students taking Core-Plus Course
1 in a given school year attend math class every other day in the fall, studying Units 1-3; these students
attend math class every day in the spring, studying Units 4-8. Because I was interested in only the first
three (algebra-focused) units of Core-Plus Course I, I conducted my research with the latter group of
students.

2“Mid-size city” as defined by the US. Census Bureau: the central city of a CMSA (Consolidated
Metropolitan Statistical Area) or MSA (Metropolitan Statistical Area) with population less than 250,000
(Indiana Department of Education, 2002).

36

The School

Oliver High School, I believe, is representative of high schools with graduating
classes of at least 300 students located in mid-size midwestem cities. To give the reader a
more elaborate yet general description of the type of school being described here, suffice
it to say that this was neither an urban nor a suburban school. That is, the River City
school corporation was characterized by neither extreme in terms of racial demographics,
academic achievement, socioeconomic standing, nor issues of safety.

For instance, the student population at Oliver was comprised of 32.5% non-white
students.3 The graduation rate was 84.2%, and the college attendance rate (those seniors
who continued their education at two- or four-year colleges or universities after high
school) was 68.0%. For college-bound seniors, the average composite SAT score was
928 (compared to the Indiana average of 1000 and the national average of 1020) with
40% of all 12th grade students at Oliver taking the SAT exam. During the year that I
collected my data, 29.4% of Oliver High School students qualified for free lunch and
another 7.3% qualified for reduced lunch. As for issues of school violence, in my
experience, there was a safe atmosphere within Oliver High School—there was a security
guard or two whose presence was nominal, school faculty and staff wore ID badges (yet
students did not), and the comings and goings of school visitors was monitored very
loosely at best. Statistically speaking, the percent of students who were suspended at
Oliver during the 2001-02 school year was 18.1%, while 7 students out of 1323 were

expelled (Indiana Department of Education, 2002).

 

3All statistical figures reflect data for the years 2000-02.

37

The Curriculum

The year when I collected my data at Oliver was the fourth year during which
Core-Plus had been used in the River City schools. From talking with several different
people within the district,4 I learned that the decision to adopt the Core-Plus curriculum
district-wide was one that was reached after much debate and difference of opinion. The
River City district superintendent had announced that reform-based mathematics
curricula would be adopted at both the middle school and high school levels, in keeping
with the statewide mathematics initiative that was underway. The superintendent
selected Oliver’s math department chairperson and two other math teachers from the
district to work for an entire school year with half-day released time to prepare for the
curricular change. Three curricula, pre-selected by the superintendent, were carefully
examined and piloted by the selection committee, and town meetings were held to allow
the public an opportunity to give input.

After the selection of Core-Plus as the new high school math curriculum, it
became district-wide policy that all 9th graders were to be automatically enrolled in
Core-Plus Course 1 if they had not already passed the course by the time they began high
school. Students first have the opportunity to enroll in Core—Plus Course 1 (for which
they receive high school credit) in 8th grade. However, those students deemed not yet
ready for this course (i.e., those who would be considered “lower track”) are enrolled in

Eighth Grade Math instead (equivalent to taking Pre-Algebra rather than Algebra I).

 

‘These informants included the River City district’s Director of Planning, Assessment & Learning
Technologies; one of the district’s high school principals (who was a middle school principal in River City
schools at the time of the Core-Plus curriculum adoption decision); an assistant principal at Oliver H.S.
(who was also a middle school principal in River City schools at the time of the Core-Plus curriculum
adoption decision); as well the math department chair and one of the math department faculty members
from Oliver High School.

38

Among all the high schools in River City, it was the math department members at
Oliver who embraced the Core-Plus curriculum most readily at the time of its adoption.
Three years after its implementation, the math faculty at Oliver seemed united in their
commitment to teaching the curriculum in the spirit in which the curriculum developers
intended (which cannot be said of all other high schools district-wide). Oliver’s math
teachers enjoyed a common lunch hour that allowed ample opportunity for collegial
support and collaboration of planning. Moreover, Oliver’s math department chair
provided teachers with unit tests and a listing of selected homework problems for use
with all students, helping ensure that everyone got a uniform curricular experience
regardless of the instructor s/he had.
The Students

The six students in my interview cohort were representative of previously
unsuccessful, white female students who come from middle- and working-class families.
Although a much more elaborate description of what is meant by “previously
unsuccessful” was given in Chapter 3, suffice it to say that these students were
representative of those girls who are one or more years behind the norm5 in taking their
first-year high school math class and whose mean overall-attitude scores fall
approximately within the middle 50% of all scores. 6 All of the students in my interview

cohort took Eighth Grade Math followed by Core-Plus Course 1 in 9th grade. Moreover,

 

sThe norm to which I refer is that which is advocated by Usiskin (1987), whereby average students take
first-year algebra in the eighth grade. This, however, is n_ot the norm at Oliver High School, where a
majority of students take Core-Plus Course 1 as ninth-graders.

6Mean overall-attitude scores can range from 1.0 (low) through 5.0 (high). Scores for students in the
interview cohort ranged from 2.33 and 3.28. Of the 31 survey-taking students, 5 had scores lower than 2.33
and 8 had scores higher than 3.28. For a detailed description of how these scores were derived, refer to the
section “Analysis of Initial Survey Data during the Data Collection Phase” later in this chapter.

39

two of the six students in the interview cohort were repeating Core-Plus Course 1 as

tenth-graders.

Data Collection
Initial Survey

In order to have a means of selecting a small cohort of students whom I could
interview throughout the course of their studies in Core-Plus Course 1A, my plan was to
administer a survey to all students enrolled in the every-other-day fall block of first-year
Core-Plus about their attitudes and behaviors related to their math studies. In addition to
attitude questions, the survey also included two problem-solving tasks: one abstract
(“naked numbers”) problem and one contextualized problem which yielded the same
solution set as the traditional problem. (This Initial Survey can be found in Appendix A.)
The classroom teachers distributed take-home letters with permission forms for parents to
sign and return in a postage-paid envelope. Of the 150 students who received these in
class, 34 returned the forms.7 From the group of students taking the survey, I planned to
select 8-12 interviewees who seemed to be previously unsuccessful in mathematics as
determined by self-reported attitudes and behaviors typically associated with non-success
in school mathematics (House, 1988; Prawat, Lanier, Byers, & Anderson, 1983) and
based on their performance on the problem-solving tasks.

On the three school days following the administration of the surveys, I observed

survey-taking students to gather supporting evidence for my preliminary categorization of

 

7Of the 34 students who returned a permission form, however, two of them never ended up taking the
survey, and the survey of a third student was eliminated from the pool after I learned that this student was
diagnosed as profoundly learning disabled and would not qualify as an interview candidate for the purposes
of the study.

40

students as previously unsuccessful or successful based on their overall attitude scores. I
took care to note students’ modes of participation in class (verbal, written, and in
collaboration within small groups) as well as any evidence of how well they seemed to be
understanding the mathematics being discussed. I was also able to collect information on
students’ race, sex, year in high school, and family structure. The latter set of
information was based in part on my classroom observations, in part on data I was able to
obtain from the school (name of guardian, race, and year in high school), and in part from
the parents’ signatures on permission forms (which offered some insight as to whether a
student lived with one or two parents or step-parents). Using all of this data, I separated
the survey-taking students into three groups: a group of 13 who I thought would be the
best interview candidates (Target Group 1); a group of 10 who I thought would be
satisfactory interview candidates (Target Group 2); and a group of 8 whom I would not
contact about the possibility of participating in the Interview Phase of my study.

The best interview candidates, for the purposes of my study, were going to be
those who seemed to have had some previous non-success in school mathematics. Thus,
those students in the group of 8 whom I did not contact for participation in my study were
those whose attitude scores on the initial survey revealed attitudes toward math that were
highly consistent with attitudes/responses expected of successful students. (An
explanation of how these scores were determined was given in Chapter 3.) Students
placed in Target Group 1 were those whose attitude scores on the initial survey revealed
attitudes toward math that were most consistent with attitudes/responses expected of
unsuccessful students. Students placed in Target Group 2 were those whose attitude

scores were not high enough to be ruled out as non-candidates for the interview phase of

41

the study, but not as low as those placed in Target Group 1. Indeed, Target Group 2
students’ scores on the attitude survey and the problem tasks revealed mixed results:
some signs of previous non-success but also some glimmers of successful traits as well.

Students’ scores on the problem tasks of the initial survey were also taken into
consideration. I was most interested in those students whose scores on the contextualized
task were especially high at the same time that their scores on the abstracted problem
were especially low. This was important to me because I had observed, when teaching
Core-Plus Course 1A myself a few years earlier, that the Core-Plus curriculum seemed to
provide previously unsuccessful Algebra One students an entrée into the subject matter—-
perhaps, I hypothesized, because of the contextualized nature of the mathematics.

From among the 23 students contacted about possible participation in the study as
interviewees, 9 female students (4 from Target Group 1 and 5 from Target Group 2)
consented to participate. It is perhaps not unusual that the interview cohort consisted of
all girls, given the fact that 21 of the 31 survey-takers were girls, and 18 of the 23
students contacted about possible participation in the study were girls. It was also not
intended that gender be an analytical category of research, although, once the cohort of
interviewees was determined, gender issues were certainly considered in the
interpretation of the research results. Nevertheless, issues particular to girls’ experiences
in school mathematics did not present themselves and are therefore not included in the

final analysis of the data.

42

Table 4.1
Summary data on interview participants as derived from initial survey8

 

 

 

 

:11: Initial

: 9., Initial Survey - Attitude Score Means9 Survey -

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N 8 Solving
g) g. Scores
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a S g .5. E. " e " ° "' g 3 °
1: _

Tem' 1 10 2nd 3 2.33 3 0 1 7 1.7 2 7 2 0 3.0 1 3 5

Andrea 1 10 2nd 110 2.44 2.3 2.7 1.3 2.7 2.3 3.3 o 4

 

Monica 1 9 lst 3 2.83 2.7 3.0 2.0 3.0 3.0 3.3 1 4

 

Amy 2 9 lst 3 2.94 2.7 2.7 1.7 3.3 4.3 3.0 l 4

 

Joy 2 9 lst 3 3.00 4.0 2.0 2.0 3.3 2.7 4.0 0 4

 

Adrienne 2 10 2nd 3 3.11 3.7 3.7 2.7 2.0 2.7 .4.0 3 4

 

Carissa 1 11 2nd 1'° 3.17 4.3 3.3 2.3 1.3 2.7 3.3 o 3.5

 

Jamie l 12 3rd 11' 3.19 3.3 3.3 3.7 2.5 2.0 3.7 O 2.5

 

Sara 2 9 lst 3 3.28 4.0 3.3 2.7 2.7 3.0 4.0 4 4

 

 

 

 

 

 

MEAN SCORES (n=3l): 3.01 3.2 3.0 2.5 2.8 3.0 3.4 0.8 3.1

 

 

 

 

 

 

 

 

 

 

 

Classroom Observations
Not only did I make classroom observations immediately following the initial

survey, but I also continued to observe interview participants in the math classroom

 

8Shaded boxes indicate scores that both (a) suggest previous non-success and (b) also fall below the
average score for all survey participants.

9A score of 3.0 is neutral. Scores below 3.0 indicate responses like those expected from unsuccessful
students; scores above 3.0 indicate responses like those expected from successful students.

"’Due to extenuating circumstances, this student was not able to meet more than one time for an interview.
” After the first interview, this student was no longer enrolled in Core-Plus Course 1 and therefore did not
continue to be part of the interview cohort.

43

setting throughout the time that interviews were being conducted.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 4.2
Summary of classroom observations of interview cohort
, g E > a E 0)
Timing of observations relatlve to -1 a B .3 cu’ 9;
Dates interviews 3' 3 Q E '9
October 4-8 Immediately following initial survey 2 2 2 l l 1
October 23-25 Beginnifiof Interview Round 1 1 l 1 l 1 0
January 24-25 Middle of Interview Round 2 1 1 l l 1 1
April 9-12 End of Interview Round 3 1 l l l 1 1
TOTAL # OBSERVATIONS: 5 5 5 4 4 3

 

 

This technique allowed me to continue collecting data on the interviewees’ modes of
participation in class and on their apparent understandings of the mathematics being

discussed. It also provided me insight into the routines and procedures of each math
class, which helped me to more accurately understand and interpret students’ reports

about their affective experiences in the classrooms of various teachers.

 

 

 

 

 

 

 

 

 

Table 4.3
Summary of interviewees’ math class enrollment
Fall (Augt to January) Spring (January through May)

Mr. Harper Mr. Larson Mr. Barnes Mr. Everett Mr. Harper
Terri Section “A” Section “F”
Monica Section “B” Section “G”
Amy Section “C” Section “H”
Joy Section “D” Section “H”
Adrienne Section “E” Section “G”
Sara Section “E” Section “J”

 

 

 

 

 

 

At the time that students were first made aware of my conducting research in the

Core-Plus Course 1A classes, they were informed that I would like to make a few

44

 

classroom visits to observe the students I would be interviewing as they participate in
class and that I might also like to note interactions that other students in class had with
interviewees. It was explained in the original information letter, which all students
received, that I would only take notes on the participation of students whose parents had
provided written consent. On the permission forms where parents gave consent for their
child to participate in the initial survey, they also had the option of giving consent for me
to take notes on their child’s participation in class.

Although all students and teachers in the classes that I visited had been made
aware that I was conducting research about the Core-Plus curriculum, the particular
identities of the participating students were kept confidential. Classroom teachers were
informed that I would not reveal to them which of their students I was observing during
my classroom visits. This was done, in part, to protect the confidentiality of participating
students. More importantly, it was done to guard against any stigmatization of
participating students, due to the fact that my study focused on students with prior
histories of non-success in school mathematics. Therefore, at no time were any students
made aware that-my study focused on previously unsuccessfiil students. Great care was
taken to let students know only that they were selected for participation as interviewees
on the basis of being able to “contribute valuable additional information to my research.”

My relationship with the classroom teachers throughout all of my visits to the
school was very open and supportive. It was made clear to the classroom teachers that I
was not there to evaluate in any way their pedagogy, but that I was there to observe
interviewees’ participation in class. Teachers readily accepted this at face value,

welcomed any questions that I had for them, and provided any supporting materials that I

45

requested (copies of tests or homework templates, seating charts, grading criteria, etc.). I
was made to feel a part of the group during the dozen or so lunch hours that I spent eating
and talking with the math department faculty on the days I was observing. Indeed,
several of the math teachers, including some whose classes were not a part of my study in
any way, bid me a fond farewell when my data collection at Oliver drew to a close.
Interviews

The second phase of my data collection was the interview phase. Between late
October and mid-April, I conducted audiotaped interviews with each of six participants
three times: once initially to obtain baseline data, and again at the conclusions of the
second and the third (algebra-focused) units of study in Core-Plus Course 1. According
to the curriculum developers, of the Course 1 units of study, the algebra content strand is
a primary emphasis in Units 2, 3, 5, 6, and 8; the functions strand is a primary emphasis
in Units 2, 3, 6, and 8. Throughout the other units of the course, connections to both the
algebra and functions content strands are included but not emphasized (Coxford et al.,
l998c, p. x). Therefore, my study focused particularly on Units 2 and 3 of the Course 1
curriculum—both of which were studied by my interview participants during the first half
of the academic year—because my interests were limited to students’ understandings of
algebraic concepts and functions that would typically be learned in a traditional first-year
algebra course. As the last lesson of Unit 1 serves as a segue between the study of
statistics and data analysis (the primary focus of Unit 1) and the study of algebra and
functions (the primary focus of Unit 2), I also chose to investigate students’ earliest
conceptions of algebra and functions by including a problem from Lesson 4 of Unit 1 in

the Post-Unit-2 Interview.

46

Each interview lasted between 30 and 60 minutes and was conducted in a small,
private room at Oliver High School at the end of the school day.12 The purpose of the
interviews was to obtain information about students’ affective experiences in Core-Plus
Course 1 as well as their developing understandings of the algebra and functions concepts
being studied. An additional purpose of the first interview was to obtain information
about students’ personal math histories. (A Summary of Instruments chart, which
provides a snapshot of the data collected, is included in Appendix A.)

At each interview session, students were asked to complete a written checklist
about their familiarity with pertinent math topics being explored in that interview’s
problem-solving tasks and a written attitude questionnaire (similar to the one given
initially). (See Familiarity Checklist in Appendix A.) After students completed the
written surveys (which typically took about 5 minutes total), they were then asked to
verbally respond to a set of questions that pertained to their experiences in math class
(which typically took 10 minutes). (See Interview Protocol A2 in Appendix A.) The
majority of the interview session was devoted to the students’ solving of three problem
tasks. The tasks were taken directly from an assessment resource booklet written by
Core-Plus curriculum authors (Coxford et al., 1998a). Each task was presented to the
student on a worksheet on which she wrote out her problem solution. Afier work on each
problem-solving task, the student was asked the same basic set of questions about her
understandings of the mathematical concepts involved in the task. (See Interview

Protocol B in Appendix A.)

 

l2This room was used during the school day by the school psychologist for testing and interviewing of
student-clients.

47

Data Analysis

Although in-depth analysis of the data was not undertaken until after completing
data collection, I did make a point to conduct some preliminary and ongoing analyses
throughout both phases of data collection, acknowledging that, when one collects
qualitative data, the “focus [of the phenomena under study] may shift as analytical
categories and theory ‘emerge’ from the data” (Laney, 1993, p. 2). Indeed, such constant
comparison methodology (Bogdan & Biklen, 1998) helped me in subsequent data
collection to tweak questions or (re-)direct the focus of my queries in response to
interesting leads that were uncovered along the way.
Analysis of Initial SurveL Dat‘a during the Daflollection Phas_e

The survey instrument was designed to be the primary means of identifying, among

the students who completed the questionnaire, those who possessed school attitudes and
behaviors that are typically associated with previous non-success in math. Survey items
were written to elicit students’ attitudes in several categories (House, 1988), as shown
below.

PUBLIC SELF-CONFIDENCE IN ONE’S OWN MATHEMATICAL
ABILITIES

I ofien volunteer to answer questions in math class.

. I understand math well enough to help a classmate who is struggling.
3. I do _n_ot_ like to go to the board to show my answer to a math
homework problem.

Nv—s

PERSONAL SELF -CONFIDENCE IN ONE’S OWN
MATHEMATICAL ABILITIES

4. I am able to tell for myself whether an answer on my math homework
is correct (or at least reasonable) without looking up the answer.
Math often does _n_o_t make sense to me.

6. I am naturally good at math.

 

5"

48

ENTHUSIASM ABOUT MATHEMATICS

7. Math was one of my favorite subjects in middle school.

8. I love playing games in math where you have to be the fastest one to
answer in order to win.

9. I can imagine myself completing a math-related major in college.

PERCEIVED SIGNIFICANCE OF MATHEMATICS

10. I want to get a job that uses as little math as possible.

11. A student doesn’t need more than 2 years of math in high school if he
(or she) is not going to go to college.

12. A student doesn’t really need to take 4 years of math in high school
unless she (or he) is going to be a math major in college.

LONG-TERM STICK-TO-IT-IVENESS IN THE STUDY OF
MATHEMATICS

13. I would be happy to get a C as my grade in Core-Plus Course 1.

14. I do not plan to take more math courses than the two that are required
to graduate from high school (Core-Plus Course 1 and Core-Plus
Course 2).

15. I want to graduate from high school with an Academic Honors
Diploma.

SHORT-TERM STICK-TO-IT-IVENESS ON MATHEMATICAL
TASKS

16. When a math problem is challenging, I usually keep working at it
until I get it figured out.
17. After finishing a math test, I usually check over my answers before
turning it in.
18. When I get back my graded math tests, I usually go over the problems
I missed so I understand why I got them wrong.
In response to each item, students were asked to indicate whether they strongly agreed
(SA), agreed (A), were undecided (U), disagreed (D), or strongly disagreed (SD) with the
statement. As survey data came in, I scored the responses to the attitude questions (1, for
the response most likely from an unsuccessful student, through 5, for the response most

likely from a successful student), as shown in the example below. (The complete Scoring

Rubric for the Attitude Questions can be found in Appendix B.)

49

Table 4.4
Example of scoring procedure for survey items

 

SA A U D SD

 

4. I am able to tell for myself whether an answer on my 5 4 3 2 1
math homework is correct (or at least reasonable)
without looking up the answer.

 

 

 

 

 

 

 

 

5. Math ofien does no; make sense to me.

 

This scoring procedure was used to calculate an overall attitude score (mean of the 18
responses), as well as specific attitude scores (mean of the 3 responses) in each of the six
subcategories: Public Self-Confidence, Personal Self-Confidence, Enthusiasm,
Significance, Long-Term Stick-to-it—iveness, and Short-Term Stick-to-it-iveness. I also
developed my own rubrics (similar to that used by Thompson & Senk, 1993) to score
students’ work on the problem-solving tasks. (See Appendix B.)

This analysis of survey data during the data collection phase was essential to
being able to proceed with the next phase of data collection, as it provided the means for
selecting those students who best qualified as interviewees according to the purposes of
the study.

Analysis of Interview Data during the Data Collection Phase

Ongoing analyses of interview data proved helpful as well. After each interview
with a participant, I wrote an interview memo. The memo served several purposes. First,
it allowed me to record data from the interview that could not be recorded on audiotape—
for example, a student’s counting on her fingers, or numbers she punches into the
calculator. These data would have been quickly forgotten and forever lost if not recorded

soon after the interview.

50

Second, the interview memo became a sort of abstract for the interview, or a
catalog of noteworthy events that took place during the interview. I anticipated there
being—at some point while doing in-depth analysis and/or writing—particular instances
from interviews that I wished to go back and examine. It would be very time-consuming
to have to listen to three hours of interviews on audiotape to locate such an instance
(assuming I was able to remember which informant was involved in the instance I seek),
yet it would be fairly simple to look back through the interview memos to locate the
desired episode.

Lastly, and perhaps most importantly, the interview memos served as a forum
where I kept track of issues that presented themselves as potentially salient. Because I
was not able, due to time constraints, to transcribe the interviews as they occurred, the
interview memos were most useful as a growing set of research notes. I made a special
effort to note unexpected happenings—such as misinterpretations of a problem task that
was posed, or delightful insights on the part of the interviewe that I hadn’t anticipated—
especially if such insights were common to more than one participant. For example,

Joy isn’t the first one who seemed not to be able to grasp the abstract idea of a

y=x line going through any (unplotted) points where y=x; others also seemed

able to focus/conceive of such points only in terms of the already-plotted points

(74, 74) and (80, 80).

(Memo of Interview Two with Joy, 1/17/02)

I also made note of particular verbal exchanges and student explanations that I wanted to
go back and examine using the transcript, once it was available.

We were pursuing the question of whether the graphs of the two trips’ expenses
were ever going to meet or cross. There is a fascinating explanation on the

51

transcript for the reasoning she came up with by looking at the hand-drawn graph
and examining & conjecturing about its downward slope! .’
(Memo of Interview Two with Sara, 1/09/02)

Likewise, I noted hunches, hypotheses, and interesting leads as they began to present

themselves.

Amy surprised me by not remembering a lot of this stuff from Unit 3 — calculating

linear regression models, writing equivalent equations, distributive property, and

so on. Both Terri and Adrienne — whom Amy has outperformed mathematically

in previous interviews — remembered these things from Unit 3 quite a bit better.

This made me wonder whether the fact that Terri and Adrienne are both taking

the course for the second time has anything to do with their remembering the stuff

better than Amy, who is taking the course for the very first time?

(Memo of Interview Three with Amy, 3/12/02)
Further Analysis of Initial Survey and Interview Data

Once all interviews were complete, the first task I undertook was to transcribe the
interview tapes. Audiotapes of 14 of the 15 interviews conducted with the six
participants were available.'3 I personally transcribed all three interviews with Adrienne
and then sought assistance in transcribing the remaining interviews, choosing instead to
spend my time working on other means of data analysis. The transcriptionists I hired
typed up only the problem-solving portions of the interviews. Meanwhile, I recorded
students’ responses to the protocol questions in charts for easy comparison of various
interviewees’ answers to similar questions.

I also scored interviewees’ responses to the attitude questionnaire items from each
interview in the same way that the attitude questionnaire items were scored from the

initial survey. Then I studied these scores, both for the initial survey and for the attitude

questionnaires given at each interview, and analyzed them for pattems—not only patterns

 

13The audiotape of Interview One with Amy was blank. It is suspected that the Play button was pushed
without also pushing Record on the cassette recorder.

52

among the responses of each interviewee but also patterns among the responses of all
interviewees as a group. In the responses to the initial survey questions, I discovered a
fascinating trend: The interviewees’ responses revealed consistently low levels of
enthusiasm for math at the same time that their responses revealed consistently high
levels of short-term stick-to-it-iveness. That is, despite their not liking math as a subject,
the interviewees nevertheless demonstrated productive learning habits in their math
studies. (Whole Group — Initial Survey Results are included in Appendix B.)

Perhaps the most significant component of data analysis was the evaluation of
students’ problem-solving work on the interview tasks. Prior to undertaking the grading
of students’ work, I carefully examined each task that was used in the interviews and
noted each mathematical concept or problem-solving skill that interested me. I then

categorized these evaluation items according to type.

 

 

 

 

 

 

 

Table 4.5
Performance Skills and their Types for Interview 1 Task 3

PERFORMANCE SKILL TYPE

Understanding that the (Eestion implies Which option is more economical? U

Recognimg that the answer to thwestion has more than one case U

Usingpaired data to comjfie two functions TS
‘U_sing an explicit formula to compare two functions RS

DerivingNOW-NEXT equations for calculatingvalues R

Using the Answer Key on the TI-83 to quickly calculate values iteratively (Any Task) C

 

 

 

 

53

Table 4.6
Definitions for Performance Skill Types

 

U Comprehension of what the problem was asking;
Understanding the nature of the problem

 

TS This item notes whether a table aided the student in obtainig the solution

 

RS This item notes whether a rule (equation) aided the student in obtaining the solution

 

W

This is a skill that supports the use of a rule (equation) to solve the problem
C Calculator skill

 

 

 

 

In order to score each student’s performance on the evaluation items for each task
of each interview, I not only found it necessary to consult the answers students had
written on their task worksheets, but I also had to consult the interview transcripts for
students’ explanations of their thinking while completing the tasks, and I consulted the
interview memos where I had written notes on each student’s work on each task during
the interview. Once the scoring of each participant’s problem-solving work was
complete, I then began to analyze the results of the evaluations both by task and by skill
type, looking for patterns in each student’s performance as well as patterns among the

various students’ problem-solving work.

Writing as Data Analysis

The data analysis described above was undertaken as a direct means of making
sense of the data. Without this kind of data analysis, one has little to write about.
Nevertheless, writing can be used as a tool of data analysis itself (Richardson, 1994), as
an indirect means of making sense of the data. This was the case in my research. In two
instances—writing portraits of the six participants in Chapter 5 and creating a map of the
algebra terrain in Chapter 3—writing/creating was used for the express purpose of further

analyzing data. In the third instance I describe below, additional insights about the data

54

 

came as an unintended, but welcome, result of writing up field notes.
Portraits of the Six Participargs

The portraits of the six interview participants presented in Chapter 5 were written
with the primary purpose of providing a purely descriptive introduction to the students
whose experiences of the Core-Plus curriculum and understandings of algebra and
functions are being explored in this study. However, a secondary purpose in writing
these portraits was to gain further insight and information about patterns that might exist
across participants described in Chapter 6. Indeed, as I looked at each student’s set of
data, with an eye toward answering each of the three questions that drive this study (What
is the nature of these students’ previous non-successes in school mathematics? What are
these students’ experiences of the Core-Plus curriculum? What do these students
understand about algebra and fimctions?), I made new discoveries and came to better
(and sometimes differently) understand the data I had.
Use of the Algebra Map

Similarly, the algebra map presented in Chapter 3 was created with the initial
purpose of providing a description of the algebra terrain that was used as a conceptual
framework in this research. The algebra map was also created with the intent to use it as
an analytic framework for helping identify what participants understood about algebra
and functions (presented in Chapter 6). However, the process of creating the map of the
algebra terrain was also an exercise in preliminary data analysis. As I collected and
organized the various fields and topics within fields that appear on the map, I was also
mentally forming new categories of analysis for later data exploration. This, in turn,

helped inform my ultimate interpretations of what participants understood about algebra

55

and functions; it also gave me a head start in identifying what participants did not
understand about algebra by virtue of the fact that certain topics appearing on the map
were not part of the Core-Plus Course 1A curriculum. While the algebra map served as a
tool for directly analyzing the data; the writing/creating of the algebra map also served as
an indirect means of making sense of the data.

The Emergence of Findings

As mentioned in a previous section of this chapter, one of my methods of keeping
track of data during the data collection process—writing a memo following each
interview—tumed out, quite unintentionally, to also be a method of preliminary data
analysis. It was in the course of recording unexpected happenings (misinterpretations or
interesting insights of students), as well as hunches, hypotheses, and interesting leads that
I first began to take note of early patterns beginning to present themselves.

It did not become apparent, however, that findings were perhaps beginning to
emerge until I began to write about my work-in-progress. The sequence of my collecting
data and drafting a dissertation proposal were a bit unconventional and, as it turned out,
the writing of my dissertation proposal took place between the second and third rounds of
interviews that I conducted with participants. As I worked to make some sense of the
data I had collected during the first two rounds of interviews in order to present the study-
in-progress to my dissertation committee, I began to see that the early patterns that had
begun to take shape seemed to hold some promise and deserved further investigation.
During the third round of interviews, then, I paid close attention to the learning issues
that seemed to underlie these patterns I was noticing, taking care to note whether

additional information would help clarify the fuzzy picture that I thought I might be

56

seeing. Indeed it did.

It is worth noting that, of these three categories of findings described in the
second part of Chapter 6, two were quite unexpected (mathematical community
membership and language issues); the third (contextualized versus abstracted
mathematics) ran parallel to—but did not confirm—a conjecture that I had in mind as I
began the study: that the contextualized nature of the Core-Plus curriculum provided
students an entrée into the subject matter that previously unsuccessful students did not
experience with traditional curricula (which typically deal far more, and often
exclusively, with abstracted mathematics). It was not my aim to confirm this conjecture.
Instead, I set it aside to allow the data to speak to me, not wanting to have any part in
trying to make the data say something I was hoping it would.

In the following two chapters, then, I present my full collection of findings.
Chapter 5 is a descriptive overview of each of the six participants individually. This first
level of analysis lays the groundwork for the broader analysis and interpretation in
Chapter 6. These second-level findings include patterns that did (or did not) exist across
participants in each of the three inquiry areas (previous non-success in math, experience
of the Core-Plus curriculum, and understanding of algebra and functions) as well as the
two categories of findings related to learning issues: mathematical community
membership and contextualized versus abstracted mathematics. These findings, in turn,
lead to the implications for high school math teachers and mathematics teacher educators

in Chapter 7.

57

CHAPTER 5

PORTRAITS OF THE SIX PARTICIPANTS

Portraits of the participants are intended to provide the reader with vivid images
of each of the six students whose learning experiences are explored in this study. As I
conducted my research with these students, the more I learned about who they were as
people, what made them unique individuals, the better I was able to understand their
learning experiences. Their stories vary quite a bit—they come from different personal
backgrounds, their previous and current school experiences differ—yet they also have a
number of elements in common. What makes these students distinct is described here in
Chapter 5. That which that they share in common is described in Chapter 6.

As I have reflected on all that I’ve come to know about these six students and
their learning experiences in high school math, I have been reminded many times of other
students—those that I myself have taught in high school math classes in the past. As I
gained insight about the things that make learning mathematics, and algebra in particular,
more or less challenging for these students, I found myself looking for ways that I might
apply these findings to my own teaching of math, and algebra in particular, should I ever
find myself back in a high school classroom again. Additionally, I sought particular
meanings for members of the community of mathematics teacher educators. These are

the implications that I share in Chapter 7.

58

Organization of the Portraits

These portraits are organized around the three questions that guide this study:
What is the nature of these students’ previous non-successes in school mathematics?
What are these students’ experiences of the Core-Plus Course 1A curriculum? What do
these students understand about algebra and functions (at three different moments in
time)? In addressing each question for each participant, an attempt at uniformity was
made, yet the priority was to offer data that would allow the reader to best understand
each participant as an individual. Data that did not offer salient information about a
participant is not included in the portrait. So, varying sets of tables (and, in some cases,
merely excerpts of data tables) appear among the portraits, for instance. Complete
versions of the data tables can be found in Appendix B. Furthermore, data that proved to
be important in the second level of analysis—looking for patterns across participants, or
that which was relevant to the three categories of findings pertaining to learning issues—
are presented in Chapter 6. The portraits are purposefully meant to be descriptive, and
not interpretive, in nature. Interpretation of findings appears in Chapter 6. Thus this
chapter and the next are meant to complement one another.

There were many ways in which the portraits could have been ordered. Because
this is a study of the ways in which students understand algebra and functions, and
because all findings ultimately have something to say about students’ mathematics
learning, I ordered the portraits according to the participants’ overall success on the series
of interview problem-solving tasks. I begin with Adrienne, whose performance was
consistently the most successful of all six participants; I conclude with Monica who

experienced the greatest difficulty in successfully completing the interview tasks.

59

Adrienne

Adrienne was a 16-year-old tenth-grader, taking Core-Plus Course 1 for the
second time. Adrienne’s parents were divorced, and she lived with her mother, and at
least one sibling (a brother a year older than she).

DistingrishingC—lmrgcteristics of Adrienne

My overall, lasting impression of Adrienne was of her extremely shy but well-
mannered nature. During my interactions with and observations of Adrienne, she almost
never spoke unless spoken to, and she didn’t make a lot of eye contact. In the classroom,
it seemed almost as if she made an effort not to be noticed among the group. It truly
seemed painful for Adrienne when all eyes were on her, as if she might not survive the
heat of the spotlight. On the other hand, Adrienne could do much better if and when the
teacher helped her one on one.

Yet at the same time, Adrienne did participate in class verbally. This was a
fascinating phenomenon that I noticed during my classroom observations: The teacher
would pose a question to the class and, in response, a number of students would voice an
answer—Adrienne’s being a regular participant. The teacher (this was actually true of
both math teachers that Adrienne had) would rarely call on an individual student; instead,
he listened to the chorus of answers, identified the answer he was seeking, and continued
with the lesson.

My initial reaction to this classroom practice—as a former high school math
teacher and now a student teacher supervisor at the university—was to think that the
discussion would be better managed if the teacher called on students who raised their

hands. I was concerned about those students who gave wrong answers that seemed

60

largely overlooked by the teacher: How would these students arrive at correct
understandings if their misunderstandings were not addressed? I wondered.

Adrienne offered a new spin on this, however. Having the freedom to speak up
without raising her hand actually provided a safe way for Adrienne to participate
verbally—something she would never do if she were required to raise her hand in order
to be individually acknowledged by the teacher. Indeed, in gathering information on
Adrienne’s previous experiences in math classes, I asked her to describe one of her “least
successful experiences in a math class, like a negative memory that may really stand out.”

Adrienne: I don’t like being called on in class and giving the wrong answer.

Interviewer: OK. Has it happened sometimes?

Adrienne: M Hm.

Interviewer: and then you just feel mortified?

Adrienne: Yeah.

(Interview One, 10/19/01)

Interestingly, Adrienne didn’t view this calling out of answers as bona fide
participation. During the first interview, I asked Adrienne whether she ever volunteered
in class in response to a question the teacher asked. Adrienne’s answer was “I don’t
volunteer—I just yell1 out the answer” (Interview One, 10/19/01). During the last
interview, I again questioned Adrienne about her classroom participation:

Interviewer: Did you participate in the class discussion today?

Adrienne: No, well, I do. We don’t raise our hands.

(Interview Three, 3/11/02)

What’s more, Adrienne seemed empowered to participate because of this non-formalized

approach to classroom discussion:

Interviewer: Think back to a time when the answer you gave in class happened to
be wrong. What happened?

 

IIt should be clarified that what Adrienne refers to as “yelling,” I considered to be speaking at a normal
conversational volume.

61

Adrienne: There’s always people yelling out answers. He just takes the right

one and writes it down.

Interviewer: So it doesn’t keep you from calling out an answer, even if you’re

not that sure.

Adrienne: Nh nh. [No.]

(Interview Three, 3/11/02)

Not only did the freedom to call out answers at will allow Adrienne the opportunity to
participate verbally in class, it also helped her understand things better. Of Mr. Larson,
Adrienne said, “When I say something wrong, he just goes through and explains it better.
I like that. He explains it better” (Interview One, 10/19/01).

In contrast to the informal and unofficial way that Adrienne participated verbally
in class, she approached her individual written work with meticulous and methodical
work habits. She worked in a very careful manner, keeping good records—either written
or through verbal commentary—of everything she did. Adrienne often had good
mathematical insights, demonstrated strong commonsense reasoning, and displayed high
levels of perseverance and curiosity in pursuing interesting questions beyond the extent
of the question asked in the given problem task. Adrienne was also quite skilled in the
use of her TI-83 calculator, more so than any of the other participants. Despite all this,
Adrienne seemed to have little awareness of how good her mathematical skills and
understandings actually were.

Adrienne’s Previous Non-Success in School Mathematics

Personal math history. Adrienne had failed Core-Plus Course 1 as a freshman. In
fact, going into her second year of high school, Adrienne did not yet have enough credits
to officially qualify for sophomore standing. One of the only pieces of information I

gathered about Adrienne’s unsuccessful first experience in Core-Plus Course 1 surfaced

when I asked her opinion of her current Core-Plus Course 1 class. She said, “It’s a small

62

class so we get more attention” (Interview One, 10/29/01). Adrienne said there were 20
in her class last year compared to 10 in her current class. She also reported that her
current teacher explained things better than her teacher the previous year. Yet even
Adrienne was rather surprised that she was doing so much better this second time through
the course.

I had the distinct impression that Adrienne seemed to believe that success in math
classes was due, at least in part, to chance or luck of the draw (getting a good teacher or a
small class). For example, in her third interview Adrienne had expressed ambivalence
about whether she was looking forward to taking Core-Plus Course 2 in the coming
school year (she checked No Opinion/Undecided). When I asked her about this, she said
her brother—who was a grade ahead of her in school—hadn’t yet passed Core-Plus
Course 1, which made her a bit hesitant/apprehensive about Core-Plus Course 2. And
this, despite the fact that she was getting an A in Mr. Harper’s class—a class that had
over 30 rambunctious students in it. While it seemed quite clear to me that she had come
a very long way in her ability to succeed in a math class, Adrienne seemed uncertain of
her own chances for success in the future.

Entering attitudes towards mathematics. Adrienne’s general feelings toward
math as a subject were surprisingly quite positive. In the initial interview, she reported
that she thought it to be “pretty easy” as well as “fun and interesting.” She even said she
had “always liked math,” that it was “fun working with the numbers.” She also reported
that she used math outside of school when trying to anticipate biweekly paycheck totals
for her fiiend, taking into account hours, overtime, taxes, etc. (Interview One, 10/29/01).

On the attitude questions of the initial survey, Adrienne’s mean response score

63

was 3.11—only slightly above a neutral score of 3.0—which indicates that her responses,
on average, were only slightly more like those responses expected of successful students
than those expected of unsuccessful students. This score placed Adrienne squarely in the
middle of the group of 31 survey-takers—one of four whose score was the median score
(2 = +0.13)—yet she had the second highest score among the six participants.

Adrienne’s survey responses identified short-term stick-to-it—iveness and both
personal and public self-confidence as her strongest suits. In each of these categories,
Adrienne’s responses were above neutral (indicating they were responses expected from
successful students) and above average for the entire group of survey-takers. Adrienne’s
personal self-confidence score was also the highest among the group of six participants.

Table 5.1
Adrienne’s scores on initial survey attitude questions by category2

 

 

 

 

 

Means within categories
Response Self- Self- Stick-to-it- Stick-to-it-
Mean confidence, confidence, Enthusiasm Significance ive-ness, ive-ness,
Public Personal Longerm Short-term
ADRIENNE 3.11 3.7 3.7 2.7 2.0 2.7 4.0

2 = +0.13 2 = +0.64 2 = +0.65 2 = +0.20 2 = -0.78 z = -0.28 z = +0.60

MEANS Q1=3 l): 3.01 3.20 2.99 2.46 2.80 2.96 3.43

StDevP: 0.752 0.722 1.042 1.050 1.033 1.057 0.951

 

 

 

 

 

 

 

 

 

 

There were three areas in which Adrienne’s responses revealed less-than-neutral
attitudes’. The first was enthusiasm for math as a subject. (It should be noted, however,
that Adrienne’s enthusiasm was above average for the group of survey-takers as a whole,
2 = +0.19, and tied for the highest Enthusiasm subscore among the six interview
participants.) Second, Adrienne was slightly less-than-neutral and slightly below average

(2 = -0.27) in her long-term stick-to-it-iveness (i.e., the desire to pursue math studies

 

2A complete table of initial survey response data can be found in Appendix B.
3On the attitude survey response scale, the middle score of 3 is neutral or undecided.

64

throughout high school and college). Third, in terms of seeing math as an important area
of study, Adrienne’s responses revealed the lowest score of all six interview participants,
which was also fairly low for the group of survey—takers as a whole (2 = -0.76).
Adrienne’s Experiences of the Core-Plus Course 1A Curriculum
Adrienne’s opinion of Core-Plus Course 1 was that it was easy, and she reported
that it got easier for her as the year progressed. Indeed, throughout the school year, she
received continually improving above-average math grades.
Adrienne reported that, since the last interview, the Core-Plus Course 1 class had
gotten “a lot easier” because she understands it so much better. She said it just
made a lot of sense, like it had not before. This definitely was borne out in the
interview/tasks. She didn’t really have an explanation for why things have come
so easily to her this year, especially in light of her having failed last year. Even
her familiarity survey shows that she didn’t understand most of the Unit 2 topics
prior to studying them this semester in Mr. Larson’s class.

(Memo of Interview Two, 1/17/02)

Table 5.2
Adrienne’s Familiarity Checklists

 

 

 

 

 

 

 

 

 

 

 

< 1—4 m u—e i-i g; t—n - t—q
. . . . 0 cr 2’: 5
Below 15 a 11st of the toprcs 1n Chapter N of 3 5," a 3%- g E g :- g 3. §‘
your math book. i 3 € 8. < 3 g 9. =1 5- .:
93": E’.‘ ‘< a» 0 o D; 2 g" 3
. ' g .~ g Q = a = a a a
For each one, mark an X 1n one of the boxes to a, a, 3 § - 9, 8 8 g
show which statement best describes how g s a g Q" 0 i" o-
. . . . . :r :r o =3 a
famrlrar you were With the topic before mg g 2 §. g g g. a
”"5 (D
semester. 8 8 g g g a g
8‘ 8‘ 5' o‘ 2‘. 0 5'
no p-o c 5 h” 0°
:3' 5" .3 "’ no 91
a a r 9 .53.
0 (D (I)
CHAPTER 1.
Stem-and-leafalots X
Number line plots X
Histograms X
Mean, median, mode X
Five-number summary X
Box plots X
Scatterplots X
Plots over time X

 

 

 

 

 

 

 

65

Table 5.2 (cont’d)

 

CHAPTER 2.
Using a y=x line X
Graphing (x,y) data X
Writing a NOW-NEXT expression X
Using the ANS key to calculate numbers X
repeatedly (TI-83)

Writing a rule usingktters

Using a rule to produce a table (TI-83)
Choosing Xmin, Xmax, Ymin, Ymax to set the
viewing window (TI-83)

Using a rule to moduce a graph (TI-83)

Using rules to produce non-linear graphs
CHAPTER 3.

Draw a line to fit the pattern in a alot

Find the linear regression model for a set of
data (TI-83)

Make predictions using linear model X
Writing equivalent equations by rearranging, X
combining2and expanding terms
Solve an equation such as 3x + 12 = 45 without X
the use of a table mph
Using a table to find values of variables that X
satisfy the conditions of two equations (TI-83)
Finding the rate of change (the slope) and y- X
intercept of a linear flit
Use the equation of a linear model to make a X
quick sketch of the graph by hand

 

 

 

 

 

 

 

 

 

 

 

 

 

 

><>< ><><><

 

 

 

><><

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Adrienne mentioned in every interview that her teachers took the time to explain
things and were very helpful when she had questions. From Adrienne’s point of view,
this was crucial to her success in class, especially because her experiences with work in
small groups (of students) did not prove to be as helpful to her as did raising her hand and
receiving help from the teacher.

During the first interview, Adrienne said that the people in her group worked
separately (side-by-side) for the most part, but that they would attempt to help one
another when someone didn’t understand; yet if they couldn’t figure it out, they would
then ask the teacher for assistance. By the second interview, Adrienne reported that she

raised her hand and asked the teacher “a lot of questions” and that the girl she was

66

usually paired with for small-group work could only help her somewhat. Then, in
Adrienne’s spring-term class, the teacher had ceased to allow students to work in small
groups because they were messing around more than they were working. Even so, the
teacher permitted students to ask a neighbor for help when needed, and he was readily
available to assist students as well.

On the written questionnaires administered at each of the three interviews,
Adrienne’s responses were consistently very positive (almost all “agree” responses and a
few “strongly agree” responses, which correspond to responses expected from successful
students). There were just six instances, in fact, where her answers changed over the
course of the semester. In the first case, Adrienne’s answer changed from negative
(Disagree) to positive (Agree) from the first to the second interview in response to the
following two statements:

- I take part in whole-group class discussions on a regular basis.

- When a math problem is challenging, I usually keep working at it until I get it

figured out.
In the second case, Adrienne’s answer to three of the questions was more strongly
positive at the end of the first term (second interview) than during the middle of either the
fall or the spring terms (first and third interviews):

I Most of the things we’re learning in Core-Plus Course 1 make sense to me.

I If I work at it hard enough, I know I can succeed in this class.

I I find it helpfiil that we use graphing calculators in this class.

In the third case, Adrienne’s answer declined from positive (Agree) to neutral

(Undecided) from the second to the third interview in response to the following question:

67

I Based on my experience in Core-Plus Course 1, I am looking forward to
taking Core-Plus Course 2.
When I asked Adrienne about her ambivalence about taking the second Core-Plus
course, she told me that her older brother was still trying to pass it as a junior, and this

caused her to question her own chances of being able to pass the course.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 5.3
Adrienne’s responses to attitude questionnaires from interview sessions
Adrienne
ATTITUDE
SUB- 29-0ct 17-Jan ll-Mar Interviews - Attitude Questions
CATEGORIES
Int 1 Int 2 Int 3
2 4 4 l. I take part in whole-group class discussions on a regular
Public Self- baSIS-
Confidence 2. I contribute my fair share of ideas during small-group
4 4 NA . . .
1nvest1gatrons.
4 5 4 3. Most of the things we’re learning in Core-Plus Course 1
Personal Self- make sense to me.
Confidence 4 5 4 4. If I work at it hard enough, I know I can succeed in this
class.
5. I am interested in the things we’re learning in Core-Plus
4 4 4
Course 1.
Enthusiasm
4 4 3 6. Based on my experience in Core-Plus Course 1, 1 am
looking forward to taking Core-Plus Course 2.
7. The problems we work on help me see the usefulness of
4 4 4 . .
Significance of math 1n everyday lrfe.
Math 4 5 4 8. I find it helpful that we use graphing calculators in this
class.
9. When a math problem is challenging, I usually keep
2 4 4 . . . .
Short-term Stick- workmg at 1t unt1l I get 1t figured out.
to-it-iveness 4 4 4 10. I am putting forth my best effort in my Core-Plus Course 1
class.
RESPONSE
MEAN: 3.6 4.3 3.9

 

 

 

 

 

 

 

68

Adrienne’s Performances on webra and Functions Tasks

Adrienne stood out among the six participants because of her consistently strong
performance on the problem-solving tasks throughout the entire study. Her score for the
contextualized problem task on the initial survey was perfect (2 = +0.73) and that for the
abstracted problem task was third highest (2 = +1.89) among the group of 31 survey-
takers. Adrienne’s performance on the problem tasks posed during the interviews also
revealed good understandings of algebra and functions. For all nine tasks, Adrienne
scored fours, based on a four-point rubric used to evaluate a student’s holistic solution for
(i.e., ultimate success on) each problem task.

Table 5.4
Adrienne’s scores on the interview problem-solving tasks

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Overall Mean Score
Solutions for all Z-score for
T1 T2 T3 Indiv'dual Grou Stats
Skills used p Gm“? 8“"
Interview 1 4 4 4 3.46 +0.20 Mean = 3.263
StDevP = 1.009
Interview 2 4 4 4 3.90 +0.64 Mean = 3.320
StDevP = 0.912
Interview 3 4 4 4 3.48 +0.44 Mean = 2.928
StDevP = 1.269
Table 5.5

Scoring Rubric for interview problem-solving tasks

 

Student did/understood this satisfactorily

 

Student eventually did/understood this with support from the interviewer

 

NW4)

Student did/understood this somewhat/partially

 

1 Student did not do/understand this satisfactorily

 

 

 

 

0 Student did not attempt this method/approach

 

A closer examination of the individual skills on each task also showed that

69

Adrienne possessed good algebra skills and understandings of algebra and functions. Her
task-skills performance means on all tasks and sub-tasks ranged from 3.00 to 4.00, where
3 indicates that the student satisfactorily completed a skill or understood a concept with
scaffolding from the interviewer, and 4 indicates that the student satisfactorily completed
a skill or understood a concept all on her own. (For a complete listing of all the task-
skills that were evaluated, see Appendix B.)

Adrienne equaled or scored higher than the mean for all six participants on all
tasks and subtasks as well. In the majority of instances, her own mean score exceeded
the group’s mean score by a noticeable amount. This is not to say that Adrienne’s
performances on these tasks was flawless, but her overall performance was the best of the
interview cohort.

Table 5.6
Adrienne’s performance means for evaluation items

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Adrienne MEAN for all
Performance Mean 6 participants
Intl Task-Skills Combined 3.46 3.25
Intl - Task 1 3.29 2.91
Intl - Task 2 4.00 3.75
Intl - Task 3 3.50 3.51
Int2 Task-Skills Combined 3.90 3.46
Int2 - Task 1 3.88 3.14
Int2 - Task 2 4.00 3.65
Int2 - Task 3a-3d 4.00 3.85
Int2 - Task 3e-3f 3.78 3.08
Int3 Task-Skills Combined 3.48 2.91
Int3 -Task 1 3.00 2.90
Int3 -Task 2 3.82 3.12
Int3 -Task 3, concrete 3.83 3.14
Int3 -Task 3, abstracted 3.00 1.88
MEAN for 3 INTERVIEWS 3.62 3.21
StDevP 0.486
Z-score for 3-lnterview Mean 0.84

 

 

 

 

7O

. Most remarkable was how well Adrienne did on the abstracted items of the third
interview’s Task 3, on which all other participants struggled (Adrienne’s mean score was
3.00 while other participants’ scores ranged from 1.00 to 2.50, with a mean of 1.65 for
those five participants). What’s more, according to Adrienne’s self-assessment of her
entering knowledge of these particular topics, these were not skills that she possessed
prior to her experience in Mr. Larson’s Core-Plus Course 1 class (see Table 5.2 above).
Of all six participants, Adrienne seemed to have the best recall of a variety of things

studied several months prior with just a bit of support on my part.

Sara

Sara was a 14-year-old ninth-grader, taking Core-Plus Course 1 for the first time.
Sara’s parents were divorced, and she lived—seemingly as an only child—with her
mother.

DistingpishinaCharacteristics of Sara

 

My overall, lasting impression of Sara was of her emotional detachment with me.
Her primary loyalties seemed to lie with a particular peer group, with whom she
“clowned” both in and outside of class. She seemed to have no concern for what her
teachers (and I) thought of her and her actions, but she did seem highly conscious of
maintaining the approval of her friends.

Sara was generally soft-spoken but very matter-of-fact with her interview
responses; it was usually hard to get her to elaborate. In fact, I would frequently offer a
summary of what I perceived Sara to be saying, and would ask if it was correct—hoping

she would say more herself—but most often she would just nod in the affirmative. (Even

7l

so, when she disagreed with my paraphrasing, she spoke up definitively.)

Sara approached her class work in much the same way: She put forth effort on
that which came easily, but had no interest in pressing farther on that which took extra
thought, hard work, or sustained effort (such as proving an answer or double-checking it
for accuracy). Sara was quite confident in her own abilities—sometimes too much so for
her successful completion of mathematics problems—and seemed satisfied that her first
try would get her by well enough. Indeed, she could do better on most math tasks with
much less effort than many other students.

Nevertheless, in my first interview with Sara, she seemed reliant on the calculator
for certainty about her calculations. Her confidence in her own mental arithmetic was not
great, and she did make some minor errors at times. It took my suggestion that she do so
to get her to use the calculator initially. When she did use it, there seemed to be a boost
in her self-confidence regarding the accuracy of her answers. Still, she didn’t persevere
with the calculator, similar to the tendency noted above; it was a tool to assist her in her
first try at an answer, but it was not seen as a tool for exploring a problem-solving
situation more deeply. Nonetheless, over time, Sara became quite adept at using the
calculator; her recall for procedures and techniques involving the TI-83 was greater than
that of most interviewees, yet this potential often went untapped because Sara simply
lacked the interest in pursuing most problems beyond the minimum required.

Sara’s minimal-effort approach got her into trouble at times when it didn’t fit with
the teachers’ expectations for her work ethic. If Sara understood the key concept early in
the lesson, she might sleep for the rest of the class period; or if Sara knew points weren’t

going to be given for written work, Sara might refilse to do it. And, when pressed to

72

conform to the teachers’ expectations, Sara acted out with stubbornness and rebellion.
Nonetheless, Sara’s academic work was good enough to earn her passing grades, which
was satisfactory to her (Question 13, Initial Survey, 10/04/01).

Sara’s Previous Non-Success in School Mathemm

Personal math history. Sara reported that, in sixth grade, she was the top math
student in her class. Consequently, as a seventh-grader, she was placed in an Eighth
Grade Math class along with other sixth- and seventh-grade students. However, she did
not experience success.

Sara struggled to understand the topics being studied in the Eighth Grade Math
class and asked to be placed into Seventh Grade Math instead, although her request was
not granted. Sara did poorly enough in Eighth Grade Math during her seventh-grade year
that she had to retake the course during her eighth-grade year. Thus, she began the Core-
Plus curriculum as a ninth-grader.

Entering attitudes towards mathematics. Sara’s general opinion of math was that
it was “okay” but “not [her] favorite subject” yet she felt she did “pretty good” in it.
When asked to describe her most successful experience in a math class, or a positive
memory that stood out, Sara explained that her knowledge of math helped her in other
classes, such as when she has to measure stock in shop class. She also said she used math
outside of school when shopping to add up the prices before paying or to figure sale
discounts. (Interview One, 10/31/01)

Compared to the entire group of 31 students who took the initial survey, Sara’s
response mean on the attitude survey placed her above the group’s response mean (2-

score = +0.35). This score ranked Sarah ninth (tied with one other student) among the 31

73

survey-takers when ranked from highest response mean (rank of 1) to lowest (rank of 31),
placing her in the upper part of the third quartile (although her response mean was

nevertheless only slightly above neutral at 3.28).

 

 

 

 

 

 

 

 

 

14--

12.. flfl u
I Q a) N

1o-- ‘33 a :g ‘3 ‘3
' N «sci

8-I

6-: ._1

4..

”L—r— '— 73—.

o- -
1.0 2.0 3.0 4.0 5.0

Figure5.1

Histogram & Overlaid Boxplot of mean scores on initial survey attitude questions,
showing Sara’s placement

Sara’s survey responses identified public self-confidence, short-term stick-to-it-iveness,
and personal self-confidence as her strongest suits. In each of these categories, Sara’s
responses were above neutral (indicating they were responses expected from successful
students) and above average for the entire group of survey-takers. The areas in which
Sara’s responses revealed below-neutral attitudes (indicating they were responses
expected from unsuccessful students) were enthusiasm for math as a subject and
significance (seeing math as an important area of study). While Sara’s score in the
significance category was below the mean for the group of survey-takers (z = -0. 13), her
enthusiasm score was actually above average for the group (2 = +0.20). Sara’s mean
score in the category of long-term stick-to—it—iveness was exactly neutral and almost

exactly the same as the mean score for the group (z = +0.04).

74

Table 5.7
Sara’s scores on initial survey attitude questions by category"

 

 

 

 

 

 

 

 

Means within categories
Response Self- Self- Stick-to-it- Stick-to-it-
Mean confidence, confidence, Enthusiasm Significance ive-ness, ive-ness,
Public Personal Long-term Short-term
SARA 3.28 4.0 3.3 2.7 2.7 3.0 4.0
z=+0.36 z=+l.10 z=+0.33 z=+0.20 z=-0.l3 z=+0.04 z=+0.60
MEANSQ=31): 3.01 3.20 2.99 2.46 2.80 2.96 3.43
StDevP: 0.752 0.722 1.042 1.050 1.033 1.057 .951

 

 

 

 

 

 

 

Sara’s Experiences of the Core-Plus Course 1A Curriculum

Sara’s initial opinion of Core-Plus Course 1 was that it was “easy” but that it
repeated a lot of the same mathematics that she had done in previous math courses. The
familiarity checklists that Sara completed at each interview seemed to support the
assertion that she was already familiar with most topics of Core-Plus Course 1A prior to
her study of them in ninth grade. Indeed, with what seemed to be minimal effort on

Sara’s part, she was receiving above-average grades in Core-Plus Course 1.

 

4Complete tables of initial survey response data can be found in Appendix B.

75

Table 5.8

Sara’s Familiarity Checklists

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Below is a list of the topics in Chapter N of g :3— g % g g “a ; g g g
yourmathbook. c": 8. 8 8. < g 5-3 3 -'-" ..:
Q ‘< a ‘< 3 8 "S B a g 3
For each one, mark an X in one of the boxes to g E 2 5!; g 5 8' E 2 6:} (an
show which statement best describes how ; :4 g fr 5‘ g 5 g g '5. a
familiar you were with the topic before th__is % g 8- g % =3 5. _9: 2
semester. ... ..4 so: 5" .3 9* 81° 8
° ° .. a e 0 e
8‘ 8‘ (E: c" o 5'
a E. g in
CHAPTER 1.
Stem-and-leaf plots X
Number line plots X
Histograms X
Mean, median, mode X
Five-number summary X
Box plots X
Scatter-plots X
Plots over time X
CHAPTER 2.
Using a y=x line X
Graphing (x, ) data X
Writing a NOW-NEXT expression X
Using the ANS key to calculate numbers X
repeatedly (TI-83)
Writing a rule using letters X
Usifi a rule to produce a table (TI-83) X
Choosing Xmin, Xmax, Ymin, Ymax to set the X
viewing window (TI—83)
UsinLa rule to produce a graph (TI-83) X
Using rules to produce non-linezflraphs X
CHAPTER 3.
Draw a line to fit the pattern in a plot X
Find the linear regression model for a set of X
data (TI-83)
Make predictions using a linear model X
Writing equivalent equations by rearranging, X
combining, and expanding terms
Solve an equation such as 3x + 12 = 45 without X
the use of a table or graph
Using a table to find values of variables that X
satisfy the conditions of two equations (TI-83)
Finding the rate of change (the slope) and y- X
intercept of a linear graph
Use the equation of a linear model to make a X

mlick sketch of the graph by hand

 

 

 

 

 

 

76

 

By the third interview, Sara’s general opinion of Core-Plus Course 1 was that it
was “boring.” The aspect she enjoyed least about the class was “all the work that we
have to do.” What she enjoyed most was the fact that her friends were in the same class
and that they clowned around together. Still, Sara continued to receive above-average
grades in the class.

On the written questionnaires administered at each of the three interviews, Sara’s
responses were largely positive (almost all “agree” responses and a few “strongly agree”
responses, which correspond to responses expected from successfill students). However,
responses to six of the ten questions were lower on the third interview questionnaire
(completed in April 2002) than they were on the first (completed in late October 2001).
Indeed, Sara’s questionnaire response mean dropped fi'om 4.3 to 3.5 from Interview One

to Interview Three.

77

Table 5.9
Sara’s responses to attitude questionnaires from interview sessions

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Sara
ATTITUDE
SUB- 31-Oct 8-Jan 9-Apr Interviews - Attitude Questions
CATEGORIES
Int 1 Int 2 Int 3
4 4 4 l. I take part in whole-group class discussions on a regular
Public Self- basis.
Confidence 4 4 4 2. I contribute my fair share of ideas during small-group
investigations.
4 4 4 3. Most of the things we’re learning in Core-Plus Course 1
Personal Self- make sense to me.
Confidence 5 4 3 4. If I work at it hard enough, I know I can succeed in this
class.
4 3 3 5. I am interested in the things we’re learning in Core-Plus
Course 1.
Enthusiasm
4 4 3 6. Based on my experience in Core-Plus Course 1, I am
looking forward to taking Core-Plus Course 2.
5 4 3 7. The problems we work on help me see the usefulness of
Significance of math in everyday life.
Math 5 4 4 8. I find it helpful that we use graphing calculators in this
class.
4 4 4 9. When a math problem is challenging, I usually keep
Short-term Stick- working at it until I get it figured out.
WIFWCDCSS 4 3 3 10. I am putting forth my best effort in my Core-Plus Course 1
class.
RESPONSE
MEAN: 4.3 3.8 3.5

 

 

 

 

 

 

 

While one cannot conclude solely on the basis of the questionnaire data that
Sara’s attitude toward her math studies declined during the course of the study, there was
a notable decline in attitude reported by Sara herself across interviews, which was
corroborated by my occasional classroom observations from October to April. For
instance, Sara was initially not included in the target group of interview participants due

to seemingly productive levels of in-class participation. Yet, in Interview Two, Sara

78

reported that she regularly slept through her math classes and got two disciplinary
referrals in two weeks’ time from her math teacher. Moreover, in Interview Three, Sara
indicated that she was not in the habit of ever opening her math book and stated that the
presence of her friends and their clowning around in class was what she enjoyed most
about her math class during the second half of the school year.
gags Performance on Algebra_and Functions Tags

Despite the fact that Sara’s attitude toward her math studies declined throughout
the study, she consistently demonstrated above-average achievement on the algebra and
functions tasks. Indeed, Sara’s scores for the two problem tasks on the initial survey
were perfect and above-average in comparison to the group of 31 survey-takers, equaled
by only one other student (contextual problem score of 4, group mean score of 3.15;
abstract problem score of 4, group mean score of 0.77). (See complete chart of Initial
Survey results on the problem tasks in Appendix B.)

Sara’s performance throughout the study on the problem tasks posed during the
interviews also revealed good understandings of algebra and functions. For eight of the
nine tasks, Sara scored fours, based on a four-point rubric used to evaluate a student’s

holistic solution for (i.e., ultimate success on) each problem task.

 

 

 

 

 

Table 5.10
Sara’s Scores on the interview problem-solving tasks
Overall Mean Score
Solutions for all Z-score for
T1 T2 T3 Individual Grou Stats
Skills used p Gm" Sta“
Interview 1 4 4 4 3.71 +0.45 Mean = 3.263
StDevP = 1.009
Interview 2 4 4 4 3.68 +0.39 Mean= 3.320
StDevP = 0.912
Interview 3 4 4 2 3.37 +0.35 Mean = 2.928
StDevP = 1.269

 

 

 

 

 

 

 

 

 

79

A closer examination of the individual skills on each task also shows that Sara
possessed good algebra skills and understandings of algebra and functions. With the
exception of the abstract portion of Task Three on Interview Three, Sara’s task-skills
performance means (for the 10 other tasks and sub-tasks) ranged from 3.25 to 4.00,
where 3 indicates that the student satisfactorily completed a skill or understood a concept
with support from me, and 4 indicates that the student satisfactorily completed a skill or
understood a concept all on her own. (For a complete listing of all the task-skills that

were evaluated, see Appendix B.)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 5.11
Sara’s performance means for evaluation items
Sara MEAN for all
Performance Mean 6 participants
Intl Task-Skills Combined 3.71 3.25
Intl - Task 1 3.75 2.91
Intl - Task 2 4.00 3.75
Intl - Task 3 3.50 3.51
Int2 Task-Skills Combined 3.68 3.46
Int2 - Task 1 3.43 3.14
Int2 - Task 2 3.80 3.65
Int2 - Task 3a-3d 3.90 3.85
Int2 - Task 3e-3f 3.56 3.08
Int3 Task-Skills Combined 3.37 2.91
Int3 -Task 1 3.25 2.90
Int3 -Task 2 3.83 3.12
Int3 -Task 3, concrete 4.00 3.14
Int3 -Task 3, abstracted 1.25 1.88
MEAN for 3 INTERVIEWS 3.59 3.21
StDevP 0.486
Z—score for 3-Interview Mean 0.78

 

 

 

 

 

80

Amy

Amy was a 15-year-old ninth-grader taking Core-Plus Course 1 for the first time.
Amy lived with both her parents and a couple of siblings.

Distinguishing Chaaacteristics of Amy

My overall, lasting impression of Amy was of her boredom with the subject of
math.

Amy’s responses to the personal math history questions confirmed what I had

already observed: She does very well in math (has always gotten As or B5), but

she does not like it as a school subject. She says she finds the subject really

“boring.” Indeed, from my in-class observations, She appears very bored.

(Memo from Interview One, 11/07/01)

Amy’s typical classroom demeanor was quiet and compliant. Indeed, one of her
teachers referred to her as the “girl in the back who doesn’t say anything” (Observation
Notes, 1/24/02). Despite her silence, one could tell that Amy participated in the class;
she smiled at the teacher’s jokes and responded when spoken to. She listened when the
teacher demonstrated something at the board and took‘notes, asked questions when
necessary, but for the most part seemed to prefer to be a silent participant in class. She
appeared to do her work in class and get it done quickly, and then have nothing else to do
that interested her.

Amv’s Previous Non-Success in School Mathew

Personal math history. Of the six participants, Amy is one whose previous school

math experiences do not reveal obvious instances of non-success. Amy reported that

throughout middle school, she had gotten As and Bs in math. Still, she was not placed in

Core-Plus Course 1 in the eighth grade as many students were.

81

It was in her 8th-grade math class, however, that Amy reported having had her
least successful experience (a negative memory that stood out). The teacher of the class,
Amy reported, “didn’t teach us anything.” Instead, the students worked on their own
from the book, and the teacher would interact if the students initiated questions.
Otherwise, he just let them work without explaining anything to them. Indeed, Amy’s
familiarity checklists differed noticeably from the other six participants’ in that she
seemed far less familiar with many topics in Chapters 1 and 2 than other participants
taking Core-Plus Course 1 for the first time. Yet, by contrast, her familiarity with

Chapter 3 topics was actually noticeably better than others’.

 

 

 

 

 

 

 

 

 

 

Table 5.12
Amy’s Familiarity Checklists
< u— m I-i u—n ” t—n 5- u—
Below is a list of the topics in Chapter N of 3 % g 2%- g E: g a. E: 5. §‘
yourmathbook. é a i g < m "a: 0’ ...:
1% =‘~< o 8 % a: a 8- ~
. '— g =9: g Q =’ a = 2 ‘” 3
For each one, mark an X 1n one of the boxes to a, a =: s- - 9, ,,, '9. no
. . :2 o to £2 a
show whlch statement best descrlbes how i :4 a g Q" g - g
familiar you were with the topic before thi_s % g ? g‘ 2 g. a
semester. .. ... i: 5" g 8 8
° ° " 3 :5. 3 3‘
8' 8 § 0‘ a: g 5'
9 5% 3 5 0% 2 "°
8 a r: c Er.
(D (D U)
CHAPTER 1.
Stem-and—leaf plots X
Number line plots X
Histograms X
Mean, median, mode X
Five-number summary X
Box plots X
Scatterplots X
Plots over time X

 

 

 

 

 

 

 

 

82

Table 5.12 (cont’d)

 

CHAPTER 2.

 

Using a y=x line X

 

Graphing (x,y) data X

 

Writing a NOW-NEXT expression X

 

Using the ANS key to calculate numbers X
repeatedly (TI-83)

 

Writing a rule using letters X

 

Using a rule to produce a table (TI-83) X

 

Choosing Xmin, Xmax, Ymin, Ymax to set the X
viewinflvindow (Tl-83)

 

Using a rule to produce a graph (TI-83) X

 

Ufilg rules to produce non-linear gr_aphs X

 

CHAPTER 3.

 

Draw a line to fit the pattern in a plot X

 

Find the linear regression model for a set of data X
(Tl-83)

 

Makgredictions using a linear model X

 

Writing equivalent equations by rearranging, X
combinipg, and expandiflterms

 

Solve an equation such as 3x + 12 = 45 without the X
use of a table or graph

 

 

Using a table to find values of variables that satisfy X
the conditions of two equations (Tl-83)

 

Finding the rate of change (the slope) and y- X
intercept of a linear graph

 

 

Use the equation of a linear model to make a quick X
sketch of the graph by hand

 

 

 

 

 

 

Entering attitudes towards mathematics. Despite Amy’s above-average grades in
past mathematics classes, her responses to the attitude questions on the initial survey
revealed some surprisingly below-average scores when compared to the entire group of
survey-takers or to the group of six participants. Amy scored below neutral (indicating
that her responses were like those expected from unsuccessful students) and below
average for the group of 31 survey-takers in three of the six categories (enthusiasm for
math, public and personal self-confidence). Amy also scored below average for the
group of 31 survey-takers in the short-term stick-to-it-iveness category (2 = -O.46),

although her mean score was neutral. Amy’s scores in three of these aforementioned

83

 

categories (enthusiasm for math, public self-confidence, and short-term stick-to-it-
iveness) were the lowest of all six participants as well.

Table 5.13
Amy’s scores on initial survey attitude questions by category

 

 

 

 

 

Means within categories
Response Self- Self- Stick-to-it- Stick-to-it-
Mean confidence, confidence, Enthusiasm Significance ive-ness, ive-ness,
Public Personal Ltfl-term Short-term
AMY 2.94 2.7 2.7 1.7 3.3 43 3.0
= -0.09 z = -O.74 z = -0.31 z = -O.76 z = +0.51 2 = +1.30 2 = -O.46
MEANS (n=3l): 3.01 3.20 2.99 2.46 2.80 2.96 3.43
StDevP: 0.752 0.722 1.042 1.050 1.033 1.057 0.951

 

 

 

 

 

 

 

 

 

 

By contrast, Amy scored above neutral (indicating that her responses were like
those expected from successful students) and above average in the other two categories
(long-term stick-to-it-iveness and seeing math as an important area of study). It must be
noted, however, that Amy’s actual responses to the questions in the significance category
were, on average, only slightly above neutral. Amy’s mean scores in both of these
categories were the highest of the six participants’ scores, yet it is Amy’s mean score in
the long-term stick-to-it-iveness category that sharply stands out.

Amy was the only one of the six participants with a score in the long-term stick-
to-it-iveness category (pursuing the study of math throughout high school and college)
that was above neutral (indicating her responses were like those expected from successful
students). Moreover, hers is the third highest score of all 31 survey-takers in this
category. Were it not for Amy’s successful disposition toward the pursuit of math
throughout high school and college, Amy’s overall response mean on the initial survey
attitude questions would not have been seemingly so “average” among the 31 survey-

takers (z = -0.09). As it was, since the majority of Amy’s responses pertaining to her

84

current school math experience were neutral or negative (i.e., consistent with responses

expected from unsuccessful students), she placed 20th among the group of 3 1 , in the

middle of the second quartile.

Table 5.14

Amy’s responses to the initial survey attitude questions

Questions pertaining to current in-school experience

Responses like those expected from successful students:

 

I would be happy to get a C as my grade in Core-Plus Course 1.

SD

 

 

I often volunteer to answer questions in math class.

 

 

 

 

 

 

Neutral regonses:

 

I understand math well enough to help a classmate who is struggling.

 

I am able to tell for myself whether an answer on my math homework is
correct (or at least reasonable) without looking up the answer.

 

 

When a math problem is challenging, I usually keep working at it until I get
it figured out.

 

I usually check over my answers on a math test before turning it in.

 

When I get back my graded math tests, I usually go over the problems I
missed so I understand why I got them wrong.

 

 

Math ofien does not make sense to me.

 

 

 

 

 

 

Responses like those expected from unsuccessful students:

 

I am naturally good at math.

 

I love playing math games in class where you have to be the fastest one to
answer in order to win.

 

I do not like to go to the board to show my answer to a math homework
problem.

SA

 

 

Math was one of my favorite sub'gcts in middle school.

 

 

 

 

 

SD

 

85

 

 

 

Table 5.14 (cont’d)
Questions pertaining to plans beyond high school

Responses like those expected from successful students:

 

I want to graduate from high school with an Academic Honors Diploma.

SA

 

A student doesn’t really need to take 4 years of math in high school unless D
she (or he) is going to be a math major in college.

 

 

 

A student doesn’t need more than 2 years of math in high school if he (or D
she) is not fling to go to college.

 

 

 

 

 

 

Neutral response:

 

 

I do po_t plan to take more math courses beyond the two that are required to U
graduate from high school (Core-Plus Courses 1 and 2).

 

 

 

 

 

 

Responses like those expected from unsuccessful students:

 

I can imagine myself completing a math-related major in college. D

 

 

I want to get a job that uses as little math as possible. A

 

 

 

 

 

 

Amy’s Experiences of the Core-Plus Course 1A Curriculum

Amy’s initial opinion of Core-Plus Course 1 was that it was “boring.” At the end

of the fall term, Amy reported feeling the very same way, although her feelings

apparently were not particular to Core-Plus Course 1, but pertained to math classes in

general:

Amy:

Interviewer:
Amy:

I don’t like math. (laugh) I never have! (laugh)

I hate math. I’ve never had very good, I’ve never had teachers
that I liked very well.

You still do really well in it, though, it seems like.

It amazes me ’cause I didn’t do very well in it last year, oh, well, I
got like a B. I’ve never had very good teachers so I just don’t like
math. Plus it’s boring.

(Interview Two, 12/18/01)

Interestingly, however, during her third interview, in the middle of the spring term, Amy

reported that it was the first time in her life that she had enjoyed a math class. She

attributed this to the teacher who, she said, made the class fun by joking around. “It’s

more interesting in [his] class, so you pay more attention, but he’s a little bit harder on

you. He, like, pushes you more” (Interview Three, 3/12/02).

86

 

 

 

Still, Amy’s attitude questionnaires from the three interviews didn’t reveal any

real changes over time. Her enthusiasm for math continued to be the category in which

her responses were lowest (most like responses expected from unsuccessful students).

She also remained quite neutral in her short-term stick-to—it-iveness, despite her

agreement with the statement “If I work hard enough at it, I know I can succeed in this

 

 

 

 

 

 

 

 

 

 

 

 

 

 

class.”
Table 5.15
Amy’s responses to attitude questionnaires from interview sessions
Amy
ATTITUDE
SUB- S-Nov l7-Dec lZ-Mar Interviews - Attitude Questions
CATEGORIES
Int 1 Int 2 Int 3
4 4 3 . I take part in whole-group class discussions on a regular
Public Self- baSIS-
Confidence 4 3 4 . I contribute my fair share of ideas during small-group
investigations.
3 4 4 . Most of the things we’re Ieaming in Core-Plus Course 1
Personal Self- make sense to me.
Confidence 4 4 4 . If I work at it hard enough, I know I can succeed in this
class.
3 3 3 . I am interested in the things we’re Ieaming in Core-Plus
Course I.
Enthusiasm
3 2 3 . Based on my experience in Core-Plus Course I, I am
looking forward to taking Core-Plus Course 2.
. The problems we work on help me see the usefiilness of
4 3 3 . .
Significance of math in everyday life.
Math 5 5 4 . I find it helpful that we use graphing calculators in this
class.
. When a math problem is challenging, I usually keep
3 3 3 . . . .
Short-term Stick- working at it until I get It figured out.
to-it-iveness 4 3 3 10. I am putting forth my best effort in my Core-Plus Course 1
class.
RESPONSE
MEAN: 3.7 3.4 3.4

 

 

 

 

 

 

87

 

Amy’s Performance on Algebra and Functions Tags

To sum up my assessment of Amy’s performance on the problem-solving tasks, I
would say that she has very strong intuitions about mathematics. Amy used shortcuts in
her first attempt at a problem (e. g., using an explicit formula rather than working
iteratively to develop a table of numbers) where others struggled to envision such a
shortcut, even when prompted. She ofien seemed to make her way easily through
problem tasks that other participants found difficult or challenging. And in several
instances, despite insisting she did not remember ever doing such problems in class, Amy
outperformed other participants afier receiving a minimal amount of coaching.

On five of the nine problem-solving tasks that were posed, Amy was successful in
completing these on her own (those with scores of 4). For the other four tasks (those with

scores of 3), Amy successfully completed them with my assistance.

 

 

 

 

 

Table 5.16
Amy’s scores on the interview problem-solving taslm
Overall Mean Score
Solutions for all Z-score for
T1 T2 T3 Individual Grou Stats
Skills used P Group Stats
Interview 1 3 4 3 3.20 -0.06 Mean = 3.263
StDevP = 1.009
Interview 2 4 4 4 3.54 +0.24 Mean = 3.320
StDevP = 0.912
Interview 3 4 3 3 3.36 +0.34 Mean = 2.928
StDevP = 1.269

 

 

 

 

 

 

 

 

 

In terms of individual skills used to solve each of the problem tasks, Amy’s mean scores
hovered near average or a bit above. Amy’s biggest challenges were in working with the

mathematical ideas in abstraction, apart from the context of the problem:

88

Interestingly, for both Parts A and C [of Item 2 on Task 2], Amy went directly to
wanting/trying to write the equation out by immediately inserting a die-value. I
asked her to stick with the X first, which she did. Yet, her first inclination was to
go straight to the application rather than stick with the abstraction. She seemed far
more at ease when working with the application (calculating a value using the
number from the die) than when working with the equation in the abstract
(writing out the equation leaving it in terms of the variable X).

(Memo of Interview Three, 3/12/02)

The same was true for Items 6 and f on Task 3 of Interview 2: Amy struggled to think
beyond the immediate context of the problem to answer theoretically, based on the

pattern that could be seen in the numbers being examined.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 5.17
Amy’s performance means for evaluation items
MEAN for all

Performance Mean Amy 6 participants
Intl Task-Skills Combined 3.20 3.25

Intl - Task 1 2.78 2.9l

Intl - Task 2 4.00 3.75

Intl - Task 3 3.75 3.51

Int2 Task-Skills Combined 3.54 3.46
Int2 - Task 1 3.38 3.14
Int2 - Task 2 4.00 3.65

Int2 - Task 3a-3d 4.00 3.85
Int2 - Task 3e-3f 2.40 3.08
Int3 Task-Skills Combined 3.36 2.91

Int3 -Task 1 3.25 2.90
Int3 -Task 2 3.40 3.12

Int3 -Task 3, concrete 4.00 3.14
Int3 -Task 3, abstracted 2.50 1.88
MEAN for 3 INTERVIEWS 3.36 3.21
StDevP 0.486
Z-score for 3-Interview Mean 0.31

 

 

 

 

 

89

Joy
Joy was a lS-year—old ninth-grader, taking Core-Plus Course 1 for the first time.

Joy lived with her father, stepmother, and four siblings (ranging in age from 5 to mid-
205) in a two-bedroom house.
Distinguishing Characteristics of Joy

I am left with two overall, lasting impressions of Joy: her amazing ability to rise
above her circumstances and her brilliantly disguised contempt for mathematics. Joy had
very difficult family relationships, yet she was a wonderfully cheerful and optimistic
person. Although she seemed to receive no familial support for her endeavors, Joy
participated in the school orchestra and was very actively involved in her church youth
group.

When I observed Joy in class, she was engaged, worked diligently, frequently
asked questions of the teacher, and displayed a clever sense of humor. She always spoke
in complete sentences and her speech was peppered with a sophisticated vocabulary.
Other students seemed to regard Joy as the one to turn to if they had questions about their
assignments in class. Indeed, she was earning above-average grades in Core-Plus Course
1. Yet, in my interviews with Joy, I was astonished to learn that she absolutely despised
mathematics and saw no use for it whatsoever.

When I pointed out to Joy that her productive classroom behaviors and her good
grades in math seemed very atypical given her strong insistence of dislike for the subject,
she attributed it to her “work ethic.” It was also Joy’s desire to attend a private college
that drove her to do her very best in school. She explained that she had a couple of
opportunities for college scholarships that were contingent on her continuing to get good

grades in high school. One of these was Indiana’s Twenty-first Century Scholars

90

program in which Joy was enrolled. The program grants tuition awards for up to four
years of college at in-state (public or private) institutions of higher education. In order to
qualify, students must enroll in the program as a seventh- or eighth-grader, must meet
income eligibility (i.e., qualify for free or reduced lunch at school), and must make a
commitment to complete high school with a GPA of 2.0 or better and not use illegal
drugs or alcohol nor commit a crime.
Jov’s Previous Non-Success in School Mathema_tic_s

Personal math history. It had not always been true that Joy had gotten good
grades in her math classes. She reported that in the sixth grade her math grades ranged
from Ds to C-plusses, and in seventh grade from D3 to As. It was not until the eightth
grade that Joy’s math grades were consistently high Bs or As. Perhaps it was Joy’s
enrolling in the Twenty-first Century Scholars program that caused the change, although
I never had the occasion to inquire. It might also have been Joy’s low grades in sixth and
seventh grade that kept her from starting Core-Plus Course 1 until the ninth grade, rather
than in the eighth grade as some students did.

Entering attitudes towards mathematics. Joy’s general opinion of math was one
of strong dislike for the subject:

I can’t stand it ’cause I see no reason why, how it’s gonna help me when I am

older. Like if I’m standing at the end of the line, and I get my groceries done, and

like they say, “You can’t have this! unless you tell me the area of this cylinder!”

(Interview One, 11/08/01)
When asked if she ever used math outside of school, Joy said, “I’m sure I do, but not that
I can think of” (Interview One, 1 1/08/01).

Joy’s responses to the attitude questions on the initial survey also revealed a very

curious mix of attitudes towards math. On the one hand, Joy was adamant in her survey

91

responses that math often does M make sense to her, that she is po_t naturally good at
math, and that math was mt one of her favorite subjects in middle school. Yet, on the
other hand, she said she was able to tell for herself whether an answer on her math
homework was correct (or at least reasonable) without looking up the answer and that she
understands math well enough to help a classmate who is struggling. She strongly
disaggeed that she wanted to get a job that uses as little math as possible and said she
could imagine herself completing a math-related major in college.

True to what I observed of Joy in class, all of her responses to questions about
classroom behaviors were consistent with those expected of successful students.
However, for most questions pertaining to plans beyond high school, Joy’s responses
were much more typical of unsuccessful students.

Table 5.18
Joy’s responses to the initial survey attitude questions

Questions pertaining to current in-school experience
Responses like those expected from successful students:

I understand math well enough to help a classmate who is struggling. A

 

 

I am able to tell for myself whether an answer on my math homework is A
correct (or at least reasonable) without lookingup the answer.

 

 

When a math problem is challenging, I usually keep working at it until I get A
it figured out.

 

I usually check over my answers on a math test before turning it in. A

 

When I get back my graded math tests, I usually go over the problems I A
missed so I understand why Ijot them wrong.

 

I often volunteer to answer questions in math class. A

 

I do n_ot like to go to the board to show my answer to a math homework D
roblem.

 

 

 

 

 

 

 

I would be happy to get a C as my grade in Core-Plus Course 1. D

 

_ Responses like those expected from unsuccessful students:
LMath often does m); make sense to me. SA
1 am naturally gold at math. SD

 

 

 

Math was one of my favorite subjects in middle school. SD

 

I love playing math games in class where you have to be the fastest one to SD
answer in order to win.

 

 

 

 

 

 

92

 

 

Table 5.18 (cont’d)

Questions pertaining to plans beyond high school

Responses like those expected from successful students:

 

I want to get a job that uses as little math as possible. SD

 

 

 

 

 

 

 

I can imagine myself completing a math-related major in college. A

 

Neutral response:

 

A student doesn’t really need to take 4 years of math in high school unless U
she (or he) is going to be a math major in college.

 

 

 

 

 

 

 

 

Responses like those expected from unsuccessful students:

 

I do pot plan to take more math courses beyond the two that are required to A
_graduate from high school (Core-Plus Courses 1 and 2).

 

A student doesn’t need more than 2 years of math in high school if he (or A
she) is not going to go to college.

 

 

 

 

 

 

 

I want to graduate from high school with an Academic Honors Diploma.

 

Not surprisingly, Joy’s overall mean score for the initial attitude survey yielded a
perfectly-neutral score of 3.00. Compared to the entire group of 31 students who took the

initial survey, Joy’s score was almost exactly the same as the group’s response mean

 

 

 

 

 

(z = -o.01).
Table 5.19
Joy’s scores on initial survey attitude questions by category
Means within categories

Response Self- Self- Stick-to-it- Stick-to-it-
Mean confidence, confidence, Enthusiasm Significance ive-ness, ive-ness,
Public Personal Long-term Short-term

JOY 3.00 4.0 2.0 2.0 3.3 2.7 4.0
2 = -0.01 z = +1.10 2 = -O.95 = -0.44 z = +0.51 = -0.28 z = +0.60

MEANS (n=3l): 3.01 3.20 2.99 2.46 2.80 2.96 3.43

StDevP: 0.752 0.722 1.042 1.050 1.033 1.057 .951

 

 

 

 

 

 

 

 

 

 

Joy’s survey responses identified public self-confidence and short-term stick-to-it-
iveness as her strongest suits. In each of these categories, Joy’s responses were above
neutral (indicating they were responses expected from successful students) and above

average for the entire group of survey-takers. In fact, in both of these categories, Joy also

93

 

 

 

scored highest out of all six participants. While Joy’s response mean in the category of
significance of math (seeing it as an important area of study) was closer to neutral, it also
was above neutral and above average for the group of 31 survey-takers as well as for the
group of six participants.

The areas in which Joy’s responses revealed below-neutral attitudes (indicating
they were responses expected from unsuccessful students) were personal self-confidence,
enthusiasm for math as a subject and long-term stick-to-it-iveness (pursuing the study of
mathematics in high school and college). These last two items are not at all surprising;
however, Joy’s lowest score comparatively, in the area of personal self—confidence
(z = -0.95), seemed quite puzzling given her recent track record in math classes.

Joy: Experiences of the Core-Plus Course 1A Curriculum

Joy’s initial opinion of Core-Plus Course 1 was that it was a lot easier than she
thought it would be: “I don’t know, I just thought it’d be harder than my eighth-grade
class, and it wasn’t” (Interview One, 11/08/01). Indeed, the topics in Chapter 1 were

nothing new to Joy for the most part, according to her familiarity checklist:

94

Table 5.20
Joy’s familiarity checklist for Chapter 1 topics

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

.. a, .. ... .... .... ....

Below is a list of the topics in Chapter N of your .53 “3’ 5 39-,- g 8‘ §' 8. 5‘ 3. 8‘

mathbook. gg 2% 3;; 5;:- 53'

;< E ~< 3 g g '5; (2: :3- '3

For each one, mark an X in one of the boxes to E ' $2. =: t3- .3 a '5,‘ '3. §

show which statement best describes how familiar 5 g § .51: g g 7’ car

you were with the topic _bgfprp gig semester. g g 8- g 2 g. g
8 8 E, :3: g .8; E
s- 3 33 E 8‘ ‘3
g- 9 .3 E 0% :37 ”I;
‘3 g ' 5'

CHAPTER 1.

Stem-and-leaf plots X

Number line plots X

Histograms X

Mean, median, mode X

Five-number summary X

Box plots X

Scatterplots X

Plots over time X

 

 

Yet this pleasant surprise didn’t seem to have any influence on Joy’s enjoyment of the
class:

Interviewer: What do you enjoy least about Core-Plus Course 1?

Joy: I don’t know, the fact that it’s a m class. (laugh)

(Interview One, 1 1/08/01)
Joy’s feelings didn’t change any throughout the school year, either. At each interview,
when asked her general opinion of the class, Joy reiterated her dislike for the subject.

On the written questionnaires administered at each of the three interviews, Joy’s
responses were generally consistent over time, exhibiting the same curious mix of
responses (fairly successful class behaviors and fairly negative attitudes towards the

subject) with the mean overall scores hovering right around neutral. There was one

question on the surveys, asked both initially as well as at each of the three interviews,

95

 

however, where Joy’s answers were not consistent over time: the question of whether or
not math makes sense to the student. On the initial survey, Joy strongly agreed with
“Math often does n_ot make sense to me,” yet in two of the three interviews, she agreed
with “Most of the things we’re Ieaming in Core-Plus Course 1 make sense to me.” (At
the second interview, Joy was undecided.)

The only truly noticeable change over time was in Joy’s involvement in small-
group work. Initially, she reported contributing her fair share despite the fact that others
in the group didn’t do much work, other than to copy down answers (without
understanding the problems) when the teacher went over them. By the end of the fall-
term, the “group” work had become a formality where students sat in their assigned
groupings but worked independently (if at all). Then, during the spring term, Joy’s new

teacher did not utilize small-group work in class.

96

Table 5.21

Joy’s responses to attitude questionnaires from interview sessions

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Joy
ATTITUDE
SUB- 8-Nov l4-Jan lZ-Apr Interviews — Attitude Questions
CATEGORIES
Int 1 Int 2 Int 3
4 4 4 . I take part in whole-group class discussions on a regular
Public Self- basrs.
Confidence 5 1 NA . I contribute my fair share of ideas during small-group
investigations.
4 3 4 . Most of the things we’re Ieaming in Core-Plus Course 1
Personal Self. make sense to me.
Confidence 5 5 4 . If I work at it hard enough, I know I can succeed in this
class.
1 1 1 . I am interested in the things we’re Ieaming in Core-Plus
Course I.
Enthusiasm
3 1 2 . Based on my experience in Core-Plus Course 1, I am
looking forward to taking Core-Plus Course 2.
1 1 l . The problems we work on help me see the usefulness of
Significance of math in everyday life.
Math 3 4 4 . I find it helpful that we use graphing calculators in this
class.
2 3 3 . When a math problem is challenging, I usually keep
Short-term Stick- working at it until I get it figured out.
to-it-iveness 5 4 4 10. I am putting forth my best effort in my Core-Plus Course 1
class.
RESPONSE
MEAN: 3.3 2.7 3.0

 

Joy’s Performance on Algebra and Functions Tasks

Perhaps my overall assessment of Joy’s performance on the problem-solving tasks

could be summed up by the following comment written in my first interview memo:

I would have to say that Joy has the strongest constructivist skills of anyone!
Basically, it comes down to the fact that Joy may not seem to start out very
strong, but by the time she is done, she has discovered/revised/constructed a
solution that is mathematically superior to the rest!

(Memo of Interview One, 11/13/01)

97

 

Joy successfully completed all but two of the nine problem-solving tasks that
were posed. Four of these seven tasks were done with my aid (those with scores of 3),

while she satisfactorily understood the other three (those with scores of 4) on her own.

 

 

 

 

 

Table 5.22
Joy’s scores on the interview problem-solving tasks
Overall Mean Score
Solutions for all Z-score for
T1 T2 T3 Individual Grou Stats
Skills used p Group Stats
Interview 1 3 4 3 3.46 +0.20 Mean = 3.263
StDevP = 1.009
Interview 2 2 4 4 3.47 +0.16 Mean = 3.320
StDevP = 0.912
Interview 3 3 l 3 2.85 -0.06 Mean = 2.928
StDevP = 1.269

 

 

 

 

 

 

 

 

 

On the two tasks where Joy did not successfully complete the problems, there was
a common theme among these tasks: Their mathematical content dealt with the most
traditional of all algebra topics—making sense of the coordinate plane and symbolically
manipulating equations to solve for an unknown. Even Joy herself, on several different
occasions, declared her struggles in these areas: During the second interview she said she
“never understood why Mr. Harper kept adding an X onto the end of problems like
Y = 2 + 5X” (1/14/02). During the third interview she mentioned her unfamiliarity with
the coordinate plane: “I understand 1_n_gs_t of the stuff I do, but some of it I don’t get. Like
that grid with the Y and the X” (4/12/02). Indeed, many of the topics having to do with

these types of traditional algebra were completely new to Joy:

98

Table 5.23
Joy’s familiarity checklists for Chapters 2 and 3

 

Below is a list of the topics in Chapter N of
your math book.

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asaqi op 01 moq Manx (pump; I
warn inoqe

2’“!!me Moml 10v Pip l [we

‘arogaq asaqi uses JaAau peq 1

For each one, mark an X in one of the boxes to
show which statement best describes how
familiar you were with the topic before this
semester.

'Jaideqa sup u;

sup Sutfipms Jaqwatuar i‘uop I

 

'waqi inoqe 911111 Baa maux
1 inq ‘aiopq asaqi uaas pen 1

 

 

 

 

 

CHAPTER 2.

 

Using a fix line X

 

Graphing (x,y) data X

 

Writing a NOW—NEXT expression X

 

Using the ANS key to calculate numbers X
repeatedly (TI-83)

 

Writing a rule using letters X

 

Using a rule to produce a table (Tl-83) X ,

 

Choosing Xmin, Xmax, Ymin, Ymax to set the X
viewing window (Tl-83)

 

Using a rule to produce a graph (TI-83) X

 

Using rules to produce non—linear graphs X

 

CHAPTER 3.

 

Draw a line to fit the pattern in a plot X

 

Find the linear regression model for a set of X
data (TI-83)

 

Make predictions using a linear model X

 

Writing equivalent equations by rearranging, X
combining, and expanding terms

 

Solve an equation such as 3x + 12 = 45 without X
the use of a table or graph

 

Using a table to find values of variables that X
satistthe conditions of two equations (Tl-83)

 

Finding the rate of change (the slope) and y- X
intercept of a linear graph

 

Use the equation of a linear model to make a X
(prick sketch of the graph by hand

 

 

 

 

 

 

 

 

On the whole, Joy’s performance on the problem tasks was average for the group
of six participants. Her mean combined-skills scores for the three interviews placed her

just above the mean for the entire group of six participants (2 = +0.10).

99

Table 5.24
Joy’s performance means for evaluation items

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

MEAN for all
Performance Mean Joy 6 participants
Intl Task-Skills Combined 3.46 3.25
Intl - Task 1 3.25 2.91
Intl - Task 2 4.00 3.75
Intl - Task 3 3.67 3.51
Int2 Task-Skills Combined 3.47 3.46
Int2 - Task 1 2.57 3.14
Int2 - Task 2 3.60 3.65
Int2 - Task 3a-3d 4.00 3.85
Int2 - Task 3e-3f 4.00 3.08
Int3 Task-Skills Combined 2.85 2.91
Int3 -Task 1 2.88 2.90
Int3 -Task 2 2.50 3.12
Int3 -Task 3, concrete 3.67 3.14
Int3 -Task 3, abstracted 2.25 1.88
MEAN for 3 INTERVIEWS 3.26 3.21
StDevP 0.486
Z—score for 3-Interview Mean 0.10

 

 

 

 

Terri

Terri was a lS-year-old sophomore, taking Core-Plus Course 1 for the second
time. She lived with her grandmother and her younger brother.
Distinggishing Characteristics of Teni

My overall, lasting impression of Terri was of her resilience and the knack she
had for chatting, almost non-stop, through our interviews. Terri was open and friendly
and exhibited a happy-go-lucky attitude about school and about life in general, although
it was clear that she was not free of struggles in either of these areas.

From the very first interview, Terri shared many details of her life with me as if

we had been close friends for a long time. She loved to share the exciting news of her

100

day, interesting anecdotes about fi'iends and family members, and updates on her
activities both in and out of school. From this, I gained tremendous insight into Terri’s
background, which served to underscore her amazing recovery from a tremendously
unsuccessful school year she had had as a freshman. Most fascinating was the mature
outlook she possessed as a result of this experience——full of healthy regret over the year
of school she squandered yet making the most of the new school year, doing things the
right way the second time around.

Terri’s Previous Non-Success in School Mathematics

Personal math history. Terri had failed Core-Plus Course 1 as a freshman, along
with English, biology, and physical education, all of which she was repeating as a
sophomore. She reported that her chronic absenteeism as a freshman was to blame for
her not passing these classes. Although she was doing rather well (receiving above-
average grades) her second time through Core-Plus Course 1, Terri described math, in
general, as her “worst subject,” saying she had never done well in it.

Entering attitudes towards mathematics. Compared to the entire group of 31
students who took the initial survey, Terri’s response mean on the attitude survey placed
her in the middle of the first quartile (z = -0.90) with the sixth lowest overall response
mean. Similarly, Terri’s mean scores in every category were below the group mean. In
four of these categories—personal self-confidence, enthusiasm for math, long-term stick-
to-it-iveness, and seeing math as an important area of study—the scores were also below
neutral (indicating they were responses expected from unsuccessfiil students). In two
categories—public self-confidence and short-term stick-to-it—iveness—the response

scores averaged out to neutral, even though they were below the mean for the group of 31

101

survey-takers. In comparison to the six participants, Terri’s mean scores were lowest in
four categories and next to lowest in two of them (public self-confidence and

significance).

 

 

 

 

 

Table 5.25
Terri’s scores on initial survey attitude questions by category
Means within categories

Response Self- Self- Stick-to-it- Stick-to-it-
Mean confidence, confidence, Enthusiasm Significance ive-ness, ive-ness,
Public Personal Long-term Short-term

TERRI 2.33 3.0 1.7 1.7 2.7 2.0 3.0
z=—O.9O z=—0.28 =—1.27 z=—O.76 z=-0.13 z=—0.91 z=—0.46

MEANS (n=3l): 3.01 3.20 2.99 2.46 2.80 2.96 3.43

StDevP: 0.752 0.722 1.042 1.050 1.033 1.057 0.951

 

 

 

 

 

 

 

 

 

 

Terri’s Experiences of the Core-Plus Course 1A Curriculum

In stunning contrast to her responses to the initial survey attitude questions,
Terri’s responses to the attitude questionnaires at each of the three interviews were
overwhelmingly positive (i.e., consistent with responses expected from successful
students) and consistently the highest of all six participants. The attitude statements used
for the interview questionnaires were written so that, in all cases, positive responses
(“Agree” or “Strongly Agree”) would be the ones expected from successful students. As
can be seen in Table 5.27, every single one of Terri’s responses across the three

interviews was positive, with the vast majority of them very positive.

102

TABLE 5.26
Terri’s responses to attitude questionnaires from interview sessions

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Terri
ATTITUDE
SUB- 25-Oct l7-Dec 6-Mar Interviews — Attitude Questions
CATEGORIES
Int 1 Int 2 Int 3
5 5 5 1. I take part in whole-group class discussions on a regular
Public Self- baSIS‘
Confidence 5 5 5 2. I contribute my fair share of ideas during small-group
investigations.
4 4 5 3. Most of the things we’re Ieaming in Core-Plus Course 1
Personal Self- make sense to me.
Confidence 5 5 5 4. If I work at it hard enough, I know I can succeed in this
- class.
4 4 5 5. I am interested in the things we’re Ieaming in Core-Plus
Course 1.
Enthusiasm
5 5 5 6. Based on my experience in Core-Plus Course 1, I am
looking forward to taking Core-Plus Course 2.
7. The problems we work on help me see the usefulness of
4 4 5 . .
Significance of math in everyday life.
Math 5 5 5 8. I find it helpful that we use graphing calculators in this
class.
9. When a math problem is challenging, I usually keep
4 4 5 . . . .
Short-term Stick- working at it until I get it figured out.
to-it—iveness 5 5 5 10. I am putting forth my best effort in my Core-Plus Course 1
class.
RESPONSE
MEAN: 4.6 4.6 5.0

 

 

 

 

Terri herself marveled at her newfound success the second time through Core-
Plus Course 1, and wasn’t quite sure how to account for the dramatic turnaround.
(Certainly her consistent attendance and applying herself to her studies made a
difference, yet she was still struggling to pass English for the second time.) Across the
three interviews, Terri attributed her success in Core-Plus Course 1 to various factors. In

the firSt interview, Terri cited the small class size (13 students), the teacher’s helpfiil

103

explanations of the material, and the fimctional nature of her small group (“They’re not
the kind of people that just want to work by themselves; they’re willing to help us”) as
factors that contributed to her success. In the second interview, Terri again mentioned
the small class size and the help that she obtained from the people in her small group as
factors that contributed to her success. However, the change of classes for the spring
term brought changes in Terri’s class size (25 students) and in the classroom teacher she
had; she also reported that the small group (now five students instead of three) was not
nearly as helpful. Still, she continued to receive above-average grades, and her attitude
scores were the highest possible at the third interview.
Terri’s Performance on Algebra and Functions Tasks

On most of the algebra and functions tasks, Terri’s performance revealed good
understandings. Six of the nine tasks were completed successfully. Terri completed two
of these tasks satisfactorily on her own (those with scores of 4), and she completed four

of them with my support from the interviewer (those with scores of 3).

 

 

 

 

 

Table 5.27
Terri’s scores on the interview problem-solving tasks
Overall Mean Score
Solutions for all Z-score for
T1 T2 T3 Individm'l Grou Stats
Skills used p cm" 5“"
Interview 1 l 3 4 3.00 -0.26 Mean = 3.263
StDevP = 1.009
Interview 2 3 4 3 3.33 +0.01 Mean = 3.320
StDevP = 0.912
Interview 3 3 2 l 2.54 -O.3l Mean = 2.928
StDevP = 1.269

 

 

 

 

 

 

 

 

 

On two of the three tasks that Terri did not complete successfully—namely Tasks

2 and 3 of Interview Three—the mathematical content involved topics that Terri reported

104

not having remembered studying in class. (Terri’s classroom teacher nevertheless

reported that students did, in fact, study these topics in class.)

Table 5.28
Terri’s familiarity checklist for Chapter 3

 

Below is a list of the topics in Chapter N of
your math book.

For each one, mark an X in one of the boxes to
show which statement best describes how
familiar you were with the topic before gig
semester.

 

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'IBQMOUIOS

sssqi op 01 moq meux ererle 1

wetp inoqe 9111!] KJsA metal

I 1nq ‘smjsq sssqi uses pen 1

1119“] IIIOQE

fluttpziue moux iou pip 1 pure
‘sJOqu essqi uses reasu pen 1

'Jeideqe sup u!

sup Supipms quwsutsr Luop 1

 

CHAPTER 3.

 

Draw a line to fit the pattern in a plot

 

Find the linear regression model for a set of
data LIT-83)

 

Make predictions usig a linear model

 

Writing equivalent equations by rearranging,
combining, and expanding terms

 

Solve an equation such as 3x + 12 = 45 without
the use of a table or graph

 

Using a table to find values of variables that
satisfy the conditions of two equations (TI-83L

 

Finding the rate of change (the slope) and y-
intercept of a linearjraph

 

Use the equation of a linear model to make a
quick sketch of the graph by hand

 

 

 

 

 

 

 

These tasks also involved the most traditional of all algebra topics—symbolically

manipulating equations to solve for an unknown and making sense of the coordinate

plane. Terri had expressed in her very first interview that algebra was hard for her. In

fact, when asked what she enjoyed least about math, “algebra” was Terri’s answer. “It’s

not an easy subject,” she said (Interview One, 10/25/01).

As for Task 1 of Interview One: Although this proved to be one of the most

challenging tasks across all interviews (based on the mean skills score across all six

105

 

participants), Terri struggled with this task more than anyone. Terri’s difficulties
stemmed primarily from the wording of the question. (See Task Sheet in Appendix A.)
Because the problem in this task did not say, “At what time are the temperatures the
same?” Terri understood the question (“When do the temperatures meet?”) to be referring
to various temperatures that were the same regardless of the time, allowing for multiple
answers. Terri had similar difficulties with the wording of the problem questions on each
of the three tasks of Interview One, but none that interfered with her success on the

problem in such a major way as it did for the first task.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 5.29
Terri’s performance means for evaluation items
MEAN for all 6
Performance Mean Terri participants
Intl Task-Skills Combined 3.00 3.25
Intl - Task 1 1.75 2.91
Intl - Task 2 3.50 3.75
Intl - Task 3 4.00 3.51
Int2 Task-Skills Combined 3.33 3.46
Int2 - Task 1 3.29 3.14
Int2 - Task 2 3.50 3.65
Int2 - Task 3a-3d 3.60 3.85
Int2 - Task 3e-3f 2.33 3.08
Int3 Task-Skills Combined 2.54 2.91
Int3 -Task 1 3.13 2.90
Int3 -Task 2 2.60 3.12
Int3 -Task 3, concrete 2.17 3.14
Int3 -Task 3, abstracted 1.00 1.88
MEAN for 3 INTERVIEWS 2.96 3.21
StDevP 0.486
Z-score for 3-Interview Mean -0.58

 

 

 

 

 

106

Monica

Monica was a 14-year-old freshman, taking Core-Plus Course 1 for the first time.
She lived with her parents, all of whom immigrated from war-tom Eastern Europe when
Monica was in fifth grade. Monica lived in a neighboring midwestem state for two years
before moving to River City for her seventh-grade year. She spoke English without an
accent. The only hint of her being a non-native English speaker was that her reading
level seemed to be below grade level—perhaps fifth or sixth grade:

[S]he had trouble pronouncing “economical” as she read the DVD problem aloud.

But I said it for her, and she repeated it perfectly. I asked her if she knew what it

meant, and she said No. I said it means “it’s a better deal” to which she said that’s

what she thought, but she wasn’t sure.

(Memo from Interview One, 10/23/01)
During her freshman year, English as a New Language was one of the classes in
Monica’s daily schedule (seeming to take the place of the English class other students
would typically take as freshmen); otherwise, she attended classes with everyone else.
Besides Monica’s having told me during her first interview that she immigrated to the
United States, there was very little indication of this during my interactions with her.
Distinguishing Characteristics of Monica

My overall lasting impression of Monica was of her silly-hearted nature. She was
talkative, always in a hurry, seemingly carefree, and a bit less mature than most ninth-
graders I’ve known. The traits she exhibited caused me to regard her more as a junior
high school student than a high-schooler. Monica also seemed hyperconscious about
wanting to fit in socially among her peers, and even though this was consistent with her

junior-high-school-like behavior, I always wondered how much influence her having

immigrated in the middle of her schooling experience had on her drive to assimilate and

107

be accepted by her friends.

Monica worked through the problem-solving tasks I gave her as quickly as she
could. It seemed far more important to her to get through as much as possible, as fast as
possible, than it was to ensure quality or accuracy of her work. In fact, I’m not sure that
Monica perceived a connection between the speed of her work and its quality/accuracy.

Nevertheless, Monica also seemed to want to please me and was interested in
knowing how I evaluated her performance on the problem tasks. Monica thanked me
wholeheartedly at the end of each interview. She told me, on more than one occasion,
how much she liked meeting with me to work on math. She begged me at the end of the
study to continue to work with her—more that school year as well as during the
following school year—because she felt that the interview sessions actually helped
improve her math performance in class.

Monica’s Previous Non-Success in School

Personal math history. Monica reported that in seventh grade she was enrolled in
“ESL remedial math” and that she received a letter grade of A, both in that class and in
her eighth-grade math class. There was nothing in her personal math history that pointed
to obvious non-success in school math. However, Monica was among those who did not
take Core-Plus Course 1 until the ninth grade, although many students in the River City
school district take the course in the eighth grade for high school credit.

Entering attitudes toward mathematics. Monica’s self-described opinion of math
was fairly positive. “I think it will really help me in the future,” she said (Interview One,
10/22/01). She very much enjoyed using a calculator in class, but disliked taking tests

and quizzes. When asked to describe one of her most successful experiences in a math

108

class, Monica recalled having passed a very hard test in the eighth grade; when was asked
to describe a least successful experience, she replied, “I don’t know.”

Based on her responses to the attitude questions on the initial survey, Monica
seemed more like a previously unsuccessful student, however. Compared to the group of
31 survey-takers, Monica’s overall response mean was at the lower end of the second
quartile; she placed 22nd of 31 when ranking overall response means from high to low.
Across the six categories of responses, Monica’s scores were below the group mean in
three categories (public self-confidence, enthusiasm for math, and short-term stick-to-it-
iveness). In two of those categories (public self-confidence and enthusiasm for math),
Monica’s scores were below neutral as well (indicating that her responses were like those
expected from unsuccessful students). In the three categories where Monica scored
above the group mean, it was only slightly so, and her mean scores were exactly neutral
in all three cases (significance of math as a field of study, long-term stick-to-it-iveness,
and personal self-confidence).

Table 5.30
Monica’s scores on initial survey attitude questions by category

 

 

 

 

 

 

 

 

Means within categories
Response Self- Self- Stick-to-it- Stick-to-it-
Mean confidence, confidence, Enthusiasm Significance ive-ness, ive-ness,
Public Personal Long-term Short-term
MONICA 2.83 2.7 3.0 2.0 3.0 3.0 3.3
z=—O.23 z=—0.74 z=+0.0l z=—O.44 z=+0.l9 z=+0.04 z=—0.ll
MEANS (n=3 1): 3.01 3.20 2.99 2.46 2.80 2.96 3.43
StDevP: 0.752 0.722 1.042 1.050 1.033 1.057 0.951

 

 

 

 

 

 

 

Monica’s Experiences of the Core-Plus Course 1A Curriculum
Monica’s attitude questionnaires from each of the interviews were quite positive.

Her responses to half the attitude questions became more positive over time; all but one

109

of the rest of her responses remained steadily positive across interviews. The one

response that became neutral, rather than positive, related to the use of calculators—

perhaps because, at the time of the third interview, Monica reported that she and her

classmates were not allowed to use calculators for most of their work (on the geometry-

focused Chapter 5).

Table 5.31

Monica’s responses to attitude questionnaires from interview sessions

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Monica
ATTITUDE
SUB- 22-Oct 24-Jan 18-Mar Interviews - Attitude Questions
CATEGORIES
Int 1 Int 2 Int 3
4 4 4 . I take part in whole-group class discussions on a regular
Public Self- basrs.
Confidence 4 4 4 . I contribute my fair share of ideas during small-group
investigations.
4 5 5 . Most of the things we’re Ieaming in Core-Plus Course 1
Personal Self- make sense to me.
Confidence 4 5 5 . If I work at it hard enough, I know I can succeed in this
class.
3 4 4 . I am interested in the things we’re Ieaming in Core-Plus
Course 1.
Enthusiasm
4 5 5 . Based on my experience in Core-Plus Course 1, I am
looking forward to taking Core-Plus Course 2.
4 4 4 . The problems we work on help me see the usefulness of
Significance of math in everyday life.
Math . I find it helpful that we use graphing calculators in this
4 5 3
class.
3 4 4 . When a math problem is challenging, I usually keep
Short-term Stick- working at it until I get it figured out.
to-it-iveness 4 4 4 10. I am putting forth my best effort in my CoresPlus Course 1
class.
RESPONSE
MEAN: 3.8 4.4 4.2

 

110

 

Monica’s experiences working in small groups with other students did not seem,
from an outsider’s (especially a teacher’s) viewpoint, to be as successful as those of the
other five participants; however, Monica did not seem negatively affected. In the first
interview, Monica described the nature of her small-group work to be four students
working individually on problems, regularly comparing answers with each other. Monica
admitted she did not always understand the answers she put down; if the majority of the
group had an answer that differed from hers, then she would “copy off of theirs.” She
reported making an attempt to get group members to explain such a problem to her, but
that this wasn’t usually successful. She also reported that her attempts to explain her
differing answers to others was usually not met with good reception. Still, Monica
seemed largely unfazed. She was pleased to be getting an above—average grade in the
class and enjoyed her self-perceived success.

Monica’s Performance on Algebra_and Functions Task_s

Monica struggled the most of all of the participants with the algebra and functions
tasks. Of the nine tasks, Monica was able to complete only one of them (Task 2 of
Interview Three) successfirlly, and that was done with my support. Among the eight
tasks with which Monica had difficulty, she understood three of them partially

successfully when judging performance on each task holistically.

lll

Table 5.32
Monica’s scores on the interview problem-solving tasks

 

 

 

 

 

 

Overall Mean Score
Solutions for all Z-score for

T1 T2 T3 g‘gl‘lzlggg: Group Stats Group Stats

Interview 1 l 3 1 2.69 -l .52 Mean = 3.263
StDevP = 1.009

Interview 2 2 2 2 2.86 -1.69 Mean = 3.320

StDevP = .912

Interview 3 1 l 1 1.85 ~1.56 Mean = 2.928
StDevP = 1.269

 

 

 

 

 

 

 

 

Monica’s scores on the individual evaluation items help Show a more complete
picture of her performance on the interview tasks. These numbers allude to the fact that
Monica was able to successfully complete parts of problems within each task. In a
couple of instances (Task 1 of Interview One and the first part of Interview Two Task 3),
her subscores were not too far below the mean for all participants.

Nevertheless, Monica lacked conceptual understanding of the mathematics of the
problem tasks. Her procedural knowledge was good in many instances—and perhaps this
was the reason behind her successful grades in class—but the interview problem-solving
tasks focused largely on probing for conceptual understanding, and Monica turned up

lacking deep and flexible understandings of the mathematics being explored.

112

Table 5.33
Monica’s performance means for evaluation items

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

MEAN for all
Performance Mean Monica 6 participants
Intl Task-Skills Combined 2.69 3.25
Intl - Task 1 2.63 2.91
Intl - Task 2 3.00 3.75
Intl - Task 3 2.67 3.51
Int2 Task-Skills Combined 2.86 3.46
Int2 - Task 1 2.33 3.14
Int2 - Task 2 3.00 3.65
Int2 - Task 3a-3d 3.57 3.85
Int2 - Task 3e-3f 2.40 3.08
Int3 Task-Skills Combined 1.85 2.91
Int3 -Task 1 1.88 2.90
Int3 -Task 2 2.56 3.12
Int3 -Task 3, concrete 1.17 3.14
Int3 -Task 3, abstracted 1.25 1.88
MEAN for 3 INTERVIEWS 2.47 3.21
StDevP 0.486
Z-score for 3-Interview Mean -l.70

 

 

 

113

 

CHAPTER 6

BROADER DATA ANALYSIS AND INTERPRETATION S

In the previous chapter, I discussed my findings about the Six participants as
individuals. These portraits were descriptive in nature and focused on characteristics that
made each participant unique unto herself. The collection of information presented in
Chapter 5 constitutes the first level of data analysis.

This chapter, then, represents the second, more in-depth level of data analysis. I
discuss my findings about the participants as a group. I take a step back, and with a
wider focus attempt to identify patterns across participants. In some instances, however,
patterns do not exist and generalizations cannot be made. These are findings as well.

My primary goal in this chapter, though, is to highlight the commonalities among
participants. Where there are commonalities to be found, I look for ways to interpret
them. Nevertheless, I take great care not to attribute causality in my interpretations. It is
far beyond the scope of this study to be able to identify causes and effects of students’
Ieaming. Instead, I look to the data to help me make sense of what these students
understand about algebra and functions and of the nature of their Ieaming experiences in
Core-Plus Course 1A.

I begin this chapter by discussing trends that exist in each of the categories that
define this study: students’ previous non-successes in school mathematics, students’
Ieaming experiences in Core-Plus Course 1A, and students’ understanding algebra and
functions. In the second part of this chapter, I discuss three categories that emerged as I

conducted my study.

114

Patterns across Participants
Characteristics of these students’ previous non-successes in school mathematics

The most basic characteristic common to all Six participants was the fact that they
did not take Core-Plus Course 1 until their freshman year in high school, despite the fact
that students in River City schools were allowed to take the course as eighth graders with
a teacher’s recommendation. Four of the participants had, in the previous year or two,
received grades of D or F in math. This was not true of Amy and Monica. Yet, in Amy’s
case, she cited her previous year’s math class as her least successful, and her self-
assessments of familiarity with topics in the first couple chapters of the Core-Plus Course
1 curriculum was noticeably lower than those of other participants taking the course for
the first time. In Monica’s case, she had taken “remedial math” for ESL students two
years previous, as She assimilated to a country, culture, and language that were brand new
to her as a fifth grader.

Another general characteristic that the participants shared was their entering
attitudes towards mathematics; in response to questions about their dispositions toward
math, these six participants’ answers were, on average, unlike those expected from
successful students. All six participants’ mean responses in the category of Enthusiasm
for mathematics were below neutral]; and four of the six scored below the group mean of
the other 25 students who completed the same attitude survey. Five of the six students
disagreed—four of them adamantly so—with the statement, “Math was one of my

favorite subjects in middle school.” (Adrienne was the exception.)

 

l A score of 3.0 is neutral. Scores below 3.0 indicate responses like those expected from unsuccessful
students; scores above 3.0 indicate responses like those expected from successful students.

115

 

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117

In the category of Long-term Stick-to-it-iveness (pursuing the study of math
throughout high school and college), five of the six participants’ response means were at
or below neutral as well as at or below the group mean of the other 25 students who
completed the same attitude survey. (Amy was obviously an outlier with a score of 4.3,
which was tied for third highest score among the entire group of 31 survey-takers.)

Another attitude that the participants shared was, by contrast, one that was
associated with being successful: Short-term Stick-to-it-iveness. In response to the
statement, “When a math problem is challenging, I usually keep working at it until I get it
figured out,” four of the six participants agreed while two (Amy and Terri) were
undecided. In response to the statement, “When I get back my graded math tests, I
usually go over the problems I missed so I understand why I got them wrong,” four of the
participants agreed while two (Amy and Monica) were undecided. Half of the
participants also agreed that “I usually check over my answers on a math test before
turning it in.” Only Terri disagreed; Amy and Monica were undecided.2
Characteristics of these students’ learning egriences in school m_atherr&tic_s

All six participants, as a group and individually, were deemed “previously
unsuccessful” in mathematics at the outset of the study on the basis of possessing
attitudes towards mathematics that were typical of unsuccessfiil students and/or having
had experiences in previous math classes that were unsuccessful. On the other hand, all
six participants reported receiving above-average grades in Core-Plus Course 1
throughout the study. So it is natural to ask whether these previously unsuccessfiil

students were realizing success.

 

2 Perhaps what Amy lacks in Stick-to-it-iveness in the short run she makes up for over the long term.

118

The answer is not as simple as it seems. In explicating what was meant by the
term “previously unsuccessful” in Chapter 3, I was careful to point out that success is not
defined solely by the grades that a student receives nor by her conceptual grasp of the
mathematics. As in Joy’s case, a student might receive grades of A or B in math, and she
might possess good understandings of the mathematics, yet she would not be considered
truly successful—according to the conceptual framework of this study—as long as she is
estranged from her mathematics studies because she experiences these as neither relevant
nor important to her life. In fact, Joy provides the best example of just how elusive the
answer can be to the question of whether a student is successful in mathematics, as
defined in this study. Only after thoroughly analyzing Joy’s Ieaming experiences in
Core-Plus Course 1 did I realize that Joy did n_ot enjoy overall success because she did
not experience satisfaction.

Interviewer: It seems like you’ve consistently, on questions like [Attitude
Questionnaire #5, 6, 7], you always check in the negative area, ‘I
strongly disagree’ or ‘I disagree.’

Joy: Yeah.

Interviewer: And yet you seem to do very well in class, like the questions that
you ask, and the things that you answer. And I walked by in the
hallway upstairs and your name is on the Honors List.

Joy: Math was the only class I got a B in [or] I would’ve been on [the
list for] Highest Honors.

Interviewer: It just is really fascinating to me that somebody who feels fairly
uninterested, or that kind of thing about math, still does so well.
Do you know why that is?

Joy: 1 mean I think it’s retarded, and I hate it all, and it’s really stupid.

Interviewer: Uh huh, and you see lots of people, I would say Ln_o_s_t people, that
causes them just to not try; they don’t turn in work, ’cause they
don’t want to, and they wouldn’t be getting a B.

Joy: Well, maybe I just have a strong work ethic or something because I
realize I hay; to do it. It’s just (pause) realizing, not in denial about
it or anything. I realize I have to do it to pass [ninth] grade. I know
it’s important to some people.

(Interview Three, 4/12/02)

119

In the same way, Amy and Sara cannot be categorized as having fundamentally

successful Ieaming experiences in Core-Plus Course 1 because of their lack of

satisfaction with their math studies. For both these girls, their grades were above average

and they possessed fairly good understandings of the mathematics, yet they expressed

boredom with and a sense of alienation from the subject.

Interviewer:

Sara:
Interviewer:

Sara:

Interviewer:
Sara:
Interviewer:
Sara:

How well are you doing in the class right now? Do you know
roughly what your grade is?

B.

Is that what you expect to get based on what you know you’re
putting into it and how much you understand?

Actually, I figured I would have a better grade ’cause I do all my
work and all of it gets given back to me with a good grade on it.
So it’s really lower than you expect?

[Nods.]

Do you have an idea why that might be?

Probably because the way I act in his class. I act like I don’t care. I
sleep almost every day. I get mouthy with him. (laugh)

(Interview Two, 1/08/02)

In Sara’s case it seemed that, if anything, this estrangement was growing over the course

of the study. Sara’s change in attitude seemed to be declining, and she reported less and

less productive in-class behaviors over time.

Interviewer:

Amy:

Interviewer:
Amy:

Interviewer:

Amy:

I saw you kind of chuckling to yourself about [Attitude
Questionnaire Item] number 6.

I don’t like math. (laugh) I never have! (laugh)

I hate math. I’ve never had very good, I’ve never had teachers
that I liked very well.

You still do really well in it, though, it seems like.

It amazes me ’cause I didn’t do very well in it last year, oh, well, I
got like a B. I’ve never had very good teachers so I just don’t
like math. Plus it’s boring.

Did your opinion of Core-Plus Course 1 as a class change any
since we last talked?

It’s still boring.

(Interview Two, 12/18/01)

120

In Amy’s case, however, there was a glimmer of hope that she might be on a more
successful trajectory than before, as she reported during the third interview in mid-March
that her satisfaction was growing: She was enjoying a math class for the first time in her
life.

Monica also cannot be categorized as having fundamentally successful Ieaming
experiences in Core-Plus Course 1; however, the reasons for this are quite different than
for the students mentioned above. If Monica’s success were judged on the basis of her
self-reports, she would seem quite successfiil: Not only did her attitude questionnaire
responses suggest consistently successful characteristics, but she also reported receiving
slightly above average grades. However, Monica’s understandings of the mathematics
were good on the surface only; that is, most of Monica’s understandings of the
mathematics were procedural at best. The evaluation of her problem-solving tasks
showed that she did not fully or deeply—i.e., conceptually (Hiebert, l986)—understand

the mathematics.

121

2. Sara has a weekend job as a waitress at her family's restaurant. She works 8 hours on Saturdays
and 4 hours on Sundays. Her pay is $2.50 per hour plus tips.

3)

b)

C)

Choosebtterstorepvemntthevafiablesdaflyincomand dailytips. Writeamlethatgives
Sara's income on Saturday as a function of hertips for that day.

Lettermrclau'lyincome},g Letterfordailytips \, 2 Rule ‘ L‘VD '

5,390

Writes rule that gives Sara's income on Sunday as a function of her tips for that day.

Rule I K)

One Saturday, Sara earned 870%) in tips. Use your rule to calculate her income for that day.
Saturdayincome (YO

The next day was Mother’s Day. which is usually a good day for tips. If Sara earned $80.00 in
tips on Mother‘s Day. how much id she earn that day?

Mother's Day income v QD

Figure 6.1
Monica’s work on Task 2 of Interview Two

 

 

a. Using 1 for income and Tfor tips, I = T + (2.50)(8) or I = T + 20

b. On Sunday, [Sara] works 4 hours so her income 1 is given by the following rule:
1= T+(2.50)(4)or1= T+10

c. Set T = 70 in the rule in part a. I = 70 + 20 = 90. [Sara]’s income that Saturday was
$90.

 

d. Mother’s Day is a Sunday so set T= 80 in the rule in part b: l = 80 + 10 = 90.

 

Figure 6.2
Suggested solutions from Coxford et al. (l998a, p. 66)

122

Compared to the list of indicators of success in the category of mathematics
achievement described in Chapter 3, Monica was lacking in all five areas (text
interpretation, explanation of thinking, mathematical abstraction, retention, and
invention). Interestingly, of these five areas in which Monica was less than successful,
text interpretation and explanation of thinking stand out because of her status as a recent
immigrant and non-native English speaker. I certainly wondered about the effects of her
ESL status on her ability to communicate effectively in the language of mathematics. In
turn, I wondered about the effects of Monica’s less-than-successful math communication
ability on her understanding of and her achievement in mathematics. (These language
issues are taken up in more depth in a later section of this chapter.)

Terri can be described very similarly to Monica. According to Terri’s self-
reports, she seemed rather successful. Her attitude questionnaire responses were the
highest of any participant across all three interviews, and Terri reported receiving slightly
above average grades throughout the school year. Yet, Terri’s understandings of the
mathematics, like Monica’s, were not well-rooted conceptually. Terri performed quite a
bit better than Monica on the problem-solving tasks, yet the limits of her understandings
revealed themselves more quickly during interviews than for other participants,
especially with respect to mathematical abstraction:

Terri did the same thing as Adrienne did: RE-LABELED the y-axis
differently for the second graph.

On one hand, I can see why: They chose their range for the y-axis based on
the Washington DC numbers (20 students, $400); but this isn’t a large enough
range for the Hawaii trip (20 students, $10,000).

On the other hand, even when I challenged Terri’s choice to re-number
(because she was struggling with that a bit) and asked whether she could use the
same numbers, she said No. And then, after she got her Hawaii points plotted, I

asked about the comparative costs of the two trips: Which one is higher?
(Hawaii) then why are the points lower than the others? (because the numbers on

123

the y-axis are different). Terri didn’t seem to see the difficulty that I perceived

with the Hawaii points looking lower/smaller on the graph when they are actually

higher. Another math-community-member thing to which Terri hasn’t been
initiated.

Other than these things, I’d say Terri’s comprehension of the mathematics of
the problem was good.

(Memo for Interview Two, 1/08/02)

I have found it very difficult to reach a definitive conclusion about Terri’s
fundamental success in Core-Plus Course 1. Terri showed fantastic improvement in
many areas, not only compared to her performance from her previous experience in Core-
Plus Course 1, but also compared to her performances at the beginning of the study.
Terri showed a dramatic change in attitude over the course of the study: Her attitude
scores went from lowest of the six participants on the initial survey to highest of all
participants at all three interviews. Terri also improved in her mathematics performances
as time went on. For instance, at Interview One, Terri abandoned her attempt to graph
any part of the solution for Task 1 (which called for the student to graph two functions
simultaneously to find their point of intersection), yet she graphed one function with
considerable success at Interview Two (Task 3). Nevertheless, even in this instance,
Terri was not completely successful; given another opportunity at Interview Three (Task
2), Terri was unable to graph two functions simultaneously (this time on the calculator).
Nonetheless, Tern' was able to compare these functions using a dual table on the TI-83,
and thus succeeded with the less abstract representation of the two.

So, my answer to the question of whether Terri enjoyed fundamentally successful
Ieaming experiences in Core-Plus Course 1 is a qualified yes. Terri is on a trajectory of

improvement, although she still has quite a ways to go—especially with problems at

higher levels of abstraction (which is an issue area that is taken up in more depth in a

124

later section of this chapter, “Contextualized versus Abstracted Mathematics”). Terri is
not enjoying the kind of unqualified success that one would wish for her to ultimately
enjoy; yet, considering where she came from (having failed Core-Plus Course 1 the
previous year) and where she is headed (having shown dramatic improvement over the
course of the study), Terri is well on her way to enjoying fundamentally successful
Ieaming experiences in mathematics.

Adrienne is the one participant who seemed to enjoy unquestioned success in all
areas. This is not to say she never had difficulty or showed no indications of fragile or
incomplete understandings of mathematics. Nonetheless, Adrienne’s understandings of
the mathematics were strong across all three interviews (overall, the best of all six
participants),3 her attitude questionnaire responses consistently indicated success, and the
grades she received were well above average throughout the school year. In addition,
the methodical and meticulous ways in which Adrienne worked on the problem-solving
tasks drew my attention to a set of behaviors, common to mathematicians, that contribute
to a person’s success in mathematics when exercised. (These issues are taken up in more
depth in a later section of this chapter called “Mathematics Community Membership”)
Especially in light of the fact that Adrienne was taking Core-Plus Course 1 for the second
time, Adrienne is the best example of a previously unsuccessful student who was
subsequently enjoying fundamentally successful learning experiences in mathematics.

Interviewer: Tell me your opinion of Core-Plus Course 1 as a class.

Adrienne: It’s a small class so we get more attention. I like that.
Interviewer: What do you enjoy most about the class?

 

3 See Table B6 in Appendix B. Appendix C contains my interview memo of Interview 2 with Adrienne;
the reader can compare Adrienne’s performance on Tasks 2 and 3 to Monica’s and Terri’s performance on
these same tasks described above.

125

Adrienne: It’s easier, not easier, but easier to learn now. He goes through
stuff, explains it better.
(Interview One, 10/29/01)

Interviewer: Did your opinion of Core-Plus Course 1 as a class change any
since the first interview?

Adrienne: I actually got easier, I think. The stuff we worked on, I understood
it a lot better than the earlier stuff.

Interviewer: Do you think it was because you already had the class once before?

Adrienne: I don’t remember doing that stuff.

Interviewer: What grade were you getting in class towards the end of the
semester?

Adrienne: A-minus.

Interviewer: Is that the grade you expected to get?
Adrienne: Not really. I [expected to get] probably a B.
Interviewer: Do you think your grade is a good representation of how well you
understood the material?
Adrienne: Yeah.
Interviewer: Do you have any ideas as to why your grade turned out higher than
what you thought it might be?
Adrienne: I understood the stuff better than I thought I would.
(Interview Two, 1/17/02)
Characteristics of these students’ understandings of algebra and functions
It has been no small task to try to answer the question, What do these students
understand about algebra and firnctions? It is important that the reader keep in mind that
my study provides a very limited amount of data to use in answering this question. Three
snapshots, in the form of interviews, were taken of each of the six participants. These
snapshots provide a glimpse of what a student knew about algebra and functions at a very
early stage in her learning; they focus only on the content of Lesson 4 of Chapter 1,
Chapter 2, and Chapter 3 of Core-Plus Course 1.
Core-Plus is an integrated, spiraling curriculum that incorporates various

mathematical strands (algebra and functions, geometry and trigonometry, statistics and

probability, and discrete mathematics) throughout all three courses of the high school

126

curriculum. So a student’s opportunities to learn about algebra and functions are spread
across the 21 chapters that comprise the three-year Core-Plus curriculum.4 Therefore,
only a brief introduction to the many topics that appear on the map of the algebra terrain

(Table 3.1) are offered in the first three chapters of Core-Plus Course 1.

 

’ Each course in the Core-Plus curriculum is comprised of seven units (chapters) that introduce the
mathematics content as well as 1 unit (chapter) at the end of each course that is described as a “capstone” to
the seven preceding units. This capstone unit is “a thematic, two-week, project-oriented activity that
enables students to pull together and apply the important modeling concepts and methods developed in the
entire course” (Coxford et al., l998d, p. 7).

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130

 

6 The numbers in the table refer to evaluation item numbers that appear in the Listings of Evaluation Items

included in Appendix B.

 

Another limitation to the snapshots of data that I have is the fact that only a brief
sampling of skills, topics, and concepts could be explored in the set of three 60-minute
interviews conducted with each participant. Approximately 45 minutes per interview was
spent working on the problem-solving tasks; so an understandably limited amount of
information could be gathered in 2 hours 15 minutes on what each participant understood
about algebra and functions.

Moreover, these snapshots were taken over time — the first in October or
November; the second in December or January; the third in March or April. Two caveats
must be noted about this. First, as I mentioned in Chapter 1, my focus is on students’
understanding (a verb in the progressive tense) algebra and functions: The emphasis is on
Ieaming and understanding as a process over time. Because the learning is ongoing and
certainly extended beyond the time frame of my study—in fact, students were just
beginning the algebra-focused Chapter 6, not included in my study, during the time when
Interview Three was being conducted—the answer to what students understand does, in
fact, change over time.

Second, it is also true that the answer to this question changes over time because
the Ieaming of new concepts is often fragile and easily dislodged. This was seen during
the first couple interviews in the form of negative transfer, when students would attempt
to (mis-)apply a calculator skill they had just learned in class, for instance, to a problem
on an interview that was related to material they had learned in a previous chapter. It also
was clear that the amount of time that had passed between students’ having worked on

certain topics and skills in class and their having to revisit them in an interview one to

131

three months later resulted in some deterioration of their facility with those skills.7

Given these qualifications, I will now attempt to summarize what the six
participants understood about algebra and functions. I will begin with a finer-grained
analysis for each area of the algebra map, although there are very few patterns to be
found that hold for all six participants. So, I will follow up with a broader set of findings
about students’ understandings of algebra and functions.

Specific areas of the algebra terrain. The map of the algebra terrain (Table 3.1)
is comprised of six broad fields of study: various representations, functions, levels of
abstraction, number relationships, means of solution, and number groups. Each of these
fields consists of several elements. In this section, I offer a fine-grained, composite
analysis of what the six participants understood about algebra in the categories of these
fields and their elements, based on the problem-solving work they did on the interview
tasks.

Various Representations. The representations that students were asked to use in
the interview problem-solving tasks were graphs (both those sketched by hand and those
derived on the TI-83 graphing calculator), tables of data (both written and those derived
on the TI-83 calculator), and explicit rules (the term used in Core-Plus for equations).
Iterative rules appeared only through students’ electing to use them on their own. Of the

nine tasks, there were only three (Interview Two, Task 3; Interview Three, Tasks 1 and 3)

 

7 This phenomenon was seen most dramatically in Interview Three, conducted in March and April. The
problem-solving tasks were taken from Chapter 3 material, which the students had begun studying in late
November and had finished studying in early January (with the break over Christmas taking place as well).
Most participants remarked during Interview Three that they did not remember doing certain things in
class, even though a survey of their teachers indicated that they had, in fact, spent class time working on
those very things.

I32

where students were actually directed in the problem to utilize more than one
representation when solving a problem. In the other six tasks, there was only one kind of
representation used (additional ones, if used, were incorporated by a student’s choice).
Thus, I deliberately chose to use the term various representations here, so as not to be
confused with multiple representations, which often implies the use of several, equivalent
representations used flexibly together (e.g., Brenner et al., 1997). The evaluation of
students’ facility with various representations on these problem-solving tasks does not
reflect their facility with multiple representations used in connection with one another.

Upon examining students’ scores on evaluation items related to each of these
various representations, it would be fair to say that most students for the most part were
able to use graphs, tables, and explicit rules effectively. (And, although none of the
interview problem-solving tasks overtly called for the use of iterative rules, there were
four students whose recent use of them in class resulted in positive transfer to Task 3 of
Interview One.) In most cases where participants’ performances were weak initially, they
showed improvement over time with each of these representations.

There were a couple of exceptions to the above analysis. The first is Terri and her
use of graphs. Terri was unable to draw a graph representation at Interview One, but did
fairly well drawing a graph representation at Interview Two. She was unable, however,
to make use of the calculator’s graphing capabilities during all three interviews; and she
struggled with understanding a graph representation in the abstract (Interview Three,
Task 1). As for her use of tables, Terri showed proficiency on the calculator at Interview
Three; she also made use of tables when they were provided for her in written form, but

she lacked the invention needed to create her own when it might have been helpful. Terri

133

also showed proficiency using rules at Interview Three in conjunction with calculator-
generated tables, but not for calculator-generated graphs:

Part E [of Task 2, Interview Two] was not as difficult as Parts A-D, since its
completion doesn’t depend on previously-derived answers. With some
prompting, Terri did fairly well with the Table solution method. And she did not
hesitate in the slightest when she interpreted what the answer “4” meant.

For the Graph solution method, however, Terri used lists. She entered the
values she had from the Table into lists and then proceeded to set up two Stat
Plots. I helped with getting the lists into the right spots for Xlist and Ylist (to
move us along so we could get to the parts where I was really interested in her
understanding, of the interpretation of the solution method), although there was
evidence that she understood this herself. At some point, she entered the two
expressions into Y1 and Y2, so that they appeared in the window with her two
Stat Plots, but this was merely coincidental (that she got the graphed fimctions
along with her plotted pairs).

(Memo for Interview Three, 3/11/02)

The second exception to the above analysis is Monica and her use of graphs,
tables, and rules. Monica did all right when it came to graphing data on a scatterplot, but
was completely unable to employ mathematical abstraction in making sense of the graph
representation. She also was unable to utilize the calculator’s graphing capabilities at
Interview Two, but demonstrated fairly good retention of such skills at Interview Three
with my support.

To begin with, Monica really didn’t recall having done this. At some point
during work on this task, she stated how much she forgot over Christmas Break,
and how long it had been since they worked on this type of stuff given that it’s
now March and they did this before mid-January.

So I walked Monica through Part B, telling her at each step what to do. When
we ultimately arrived at a prediction of 363.8 meters, Monica was convinced that
this was the best answer, giving a lot of credit to the authority of the calculator.
She was also impressed with herself that the answer was so close to the one that
she had given (369 m).

I should also note that, when Monica wanted to view the scatterplot/graph of
the data she had entered into Lists 1 and 2 (stories, meters), I told her she would
have to set the StatPlot. Yet, she seemed completely unfamiliar with how to do
this. Since Plot 1 was already set up for (Type) scatterplot and for Xlist:L1,
YlistzL2 -- which was coincidentally just what she needed -- I just told her all she

I34

needed to do was turn the plot On, choosing not to linger on the mechanics of this
with her.

I did much less guiding on Part C; Monica seemed to remember, for the most
part, what to do after having done Part B. She did show some insight that others
prior to her had not -- realizing that she only needed to re-enter List 1 data and
could reuse List 2 data. (Perhaps this was aided by my having starred the
columns in which we were interested for each part of the problem.) She also
realized that she didn’t need to change the Window setting for Ymin, Ymax, and
Yscl (although I hadn’t realized it).

(Memo for Task 1 of Interview Three, 3/19/02)

Monica also struggled with tables, unless they were provided for her in written form
(indicating little, if any, invention on her part); again, she did all right with tables at
Interview Three with support from me.

I paraphrased a great deal in explaining Part E and proceeded to prompt

Monica through the Table Method [using the graphing calculator], although she

really didn’t seem to recall having done this.8 Still, once she had Jerome’s and

Lin’s equations entered as Y1 and Y2, Monica was able to comfortably find the

answer of X=4 as the value that would cause Jerome and Lin to both score 48

points.

(Memo for Task 2 of Interview Three, 3/20/02)

Lastly, when it came to using rules, Monica struggled as well, unless the equations were
provided for her.

Means of Solution. The means of solution that students were asked to use in the
interview problem-solving tasks were algorithmic (solving single equations of several
types), graphical (sketching points and/or lines), and technology-assisted means of
solution in several forms (graphing scatterplots, lines, and using linear models; rules-
generated tables; and conjecturing and investigating using the T I-83). There were also

several instances where students demonstrated invention, making successful use of

alternate means of solution.

 

8 The transcript for this portion of Interview Three with Monica is included in Appendix C.

135

On problems that called for solution by means of solving single equations, there

were four types of equations involved.

Table 6.4
Four types of equations appearing in the interview problem-solving tasks
Type I. Y = 250(8) + X Given X, solve for Y.
Type 11. Y = 1(2X) + 4(X+4) + 4X Given X, solve for Y.
Type III. Y = 11X + 12 Given Y, solve for X.
Type IV. 8 + 10X = 12 + 9X Solve for X.

Type I appeared in Interview Two Task 2, and the other three types were part of
Interview Three Task 2. The equations of Type I and II were presented within context,
while the equations of Type III and IV were presented as naked numbers. Overall, all six
participants were able to solve equations of Types 1, II, and III rather successfully; only
Sara was able to solve an equation of Type IV, which she did without hesitation or error.
On problems which called for solution by sketching points and/or lines on a
graph, all six participants were ultimately successful, showing improvement over time
from Interview One to Interview Two. Three of the six participants (Amy, Terri,
Monica) struggled at both Interviews One and Two, however, when asked to sketch a
graph to find a point of intersection of two functions; yet Adrienne, who didn’t even
attempt the sketched-graph method at Interview One, was successful in sketching two

sets of points of the same set of axes at Interview Two with my assistance.9

 

9 The transcript for this portion of Interview Two with Adrienne is included in Appendix C.

136

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137

On problems that called for solution by using the TI-83 calculator to generate
graphical representations, all six participants were able to successfully make a scatterplot
from paired data (Interviews Two and Three). All six participants were also able to
successfully use the graphing calculator to find a linear regression line for sets of paired
data (Interview Three). However, on those tasks which called for graphing functions
(linear and non-linear), all participants except Adrienne struggled initially (Interview
Two); ultimately, Sara and Amy were able to do so as well, but Joy, Terri, and Monica
were not successful in using the calculator to graph functions (Interview Three).

Table 6.5
Performance scores for participants’ using rules to graph functionsIO

 

 

 

 

 

 

 

 

 

 

 

 

 

Adrn Sara Amy Joy Terri Mon
Interview 2 Graphed one function on calculator by 4 0 l 0 O 0
Task 3e entering rule as Y=
Interview 2 Graphed two functions on calculator O 0 O 0 0 0
Task 3e by entering rules as Y=
Interview 3 Graphed two functions on calculator 4 3 4 O 0 1
Task 2.e2 by entering rules as Y=

 

On problems which called for solution by using the TI-83 calculator to generate
tables, all six participants were ultimately successful (Interview Three). This is very
interesting when one considers that the generation of a table and the generation of a
function on the TI-83 calculator both require the entry of a “Y=” rule; the only difference

between obtaining a table and obtaining a graph is in the window that one views. All six

 

'0 Score interpretation is as follows: “4” indicates that the student did or understood the item satisfactorily;
“3” indicates that the student eventually did or understood this with support from the interviewer; “2”
indicates that the student did or understood this somewhat or partially; “1” indicates that the student did not
do or understand this satisfactorily; and “0” indicates that the student did not attempt this method or
approach.

138

participants were far more adept with the rule-generated table on the TI-83 than they
were with graphing functions, especially when it came to finding a point of intersection
of two functions (Interview One). It is interesting to note that, similarly, four of six
participants were more adept with the hand-written table than they were with a sketched
graph when it came to finding a point of intersection of two functions at Interview One.

Although technology—assisted conjecture and investigation as a means of solution
were not included as part of any of the tasks, there was one instance where I improvised
and extended my line of questioning on Interview Two Task 3. This occurred during the
third participant’s interview (that is, two participants had already completed their
interviews and were not asked this set of questions). However, of the four participants
(Adrienne, Sara, Joy, Monica) whom I pressed to explore the problem further, three
(Adrienne, Sara, Joy) were able to make a reasonable conjecture and two of those
(Adrienne, Joy) were able to successfully pursue the answer using the TI-83 calculator
and its rule-generated table, comparing the Y-values of the two inverse-variation
functions as X approached positive infinity.

As for student-invented means of solution, or student-selected alternate means of
solution, the results were varied. Because no clear patterns are evident, here it is difficult
to generalize across participants. There are two interesting things of which to take note,
however. First, Joy stood out as one who was rather gified in the area of invention.
There was at least one instance during every interview where I was surprised by the
method Joy selected and by her success in obtaining a solution utilizing that method. For
instance, Joy was the only participant who successfully (and perfectly) completed

Interview One Task 1 by graphing the two functions simultaneously on one graph to find

139

their point of intersection; she did not even find it necessary to write out a table of paired
data to do so! On the other hand, there also seemed to be at least one instance during
each interview where I was surprised to see Joy struggle with concepts that other
participants handled quite easily. In at least one case—solving a Type III equation at
Interview Three—Joy compensated for her inability to successfully employ the
algorithmic method of solution by using her own method of guess and check with

complete success.

Joy: Eleven X plus twelve, ummrn, well, yeah. (pause) Mmm, how
about (pause) one (long pause while student thinks)

Interviewer: In Mr. Harper's class when you guys worked on these kinds of
problems

Joy: M hm.

Interviewer: you made this little diagram that went like this."

I" 4:“

Joy: M hm.

Interviewer: Do you remember that? Does that help you?

Joy: Yeah, I

Interviewer: Started with the X here and said what's the first thing that happens
to X?

Joy: Umm, it's multiplied

Interviewer: Multiplied by, uh huh, so multiplied by eleven.

Joy: Yeah, but don't you do that opposite?

Interviewer: And, oh yeah, I was goin', you're right. (laughs) Yes, divide by
eleven, then what happens?

Joy: Minus twelve.

Interviewer: Yeah. Oops, wrote the twelve way over there. And that equals, do
you write the forty-five here, I think? (Interviewer is not sure how
to draw the circle diagram.)

Joy: (laughs) Don't know.

Interviewer: Does that help at all?

Joy: It would if I knew ~ (laughs)

 

" This is the interviewer’s (incorrect) attempt to recreate the “circle diagram” that students were taught to
use to solve equations of Type 111. See Figure 8.1 in Appendix B or the correct circle diagram.

140

Interviewer:
Joy:
Interviewer:
Joy:
Interviewer:
Joy:
Interviewer:
Joy:
Interviewer:

Joy:
Interviewer:
Joy:
Interviewer:
Joy:
Interviewer:
Joy:
Interviewer:
Joy:
Interviewer:
Joy:
Interviewer:
Joy:
Interviewer:
Joy:

Interviewer:
Joy:
Interviewer:
Joy:
Interviewer:

(laughs) Okay.

Um. But how do I divide this? (laughs)

Then I think how it worked, you go backwards.

Oh.

You worked it this other way.

So you take forty-five minus twelve.

That's right.

Okay. (laughs)

I think that's how that worked. Anyway, it's not helping you any
more than it's helping me apparently. (laughs)
(pause) It’s like thirty-six, isn't it?

Thirty-three.

Thirty-three?

Right. Five minus two, is a three.

Oh yeah.

Four minus one is a three.

Okay. Then thirty-three divided by eleven is three.
Right.

So X equals three.

Okay, so write that down.

I 'd actually come up with that ~

Yeah.

in my head

How'd ya do it?

Um, I was thinking, I knew that, like, eleven times whatever the
number equals, you know, twenty-two or thirty three
Okay, okay.

and I was adding twelve to it.

So you were just kind of guessing and checking
Yeah.

for something here that would make that whole thing work. Okay.

(Interview Three, 4/12/02)

The second item of interest to note is that, across interviews, tables became the

solution method of choice, when participants were left to their own choosing, and their
skills in using tables improved over time. Initially (Interview One), just two of the four
participants (Adrienne and Sara) recorded data in a format resembling a table, and that
was with some aid from me. At Interview Two, four of the participants (Adrienne, Sara,
Amy, Joy) initiated the use of a table when no solution method was suggested

(demonstrating invention); Adrienne used the calculator to create a rule-generated table

141

for herself, while the other three wrote out their own tables of data on paper (Task 3). At
Interview Three, participants were asked in Task 2 to solve the problem utilizing a
variety of methods and then indicate the method they preferred. All six participants
utilized the table as a means of solution most successfully, and all six participants named
the table as their preferred method of solution.

Levels of Abstr3action. There were four levels of abstraction that students

 

encountered on the interview problem-solving tasks. These are listed and briefly

described in Table 6.5.

Table 6.6
Levels of Abstraction encountered in the interview problem-solving tasks

Level I. Using math to make sense of a situation
Organizing information, describing patterns or relationships, predicting outcomes

Level II. Mathematizing a situation
Using iterative Now-Next equations, translating words into symbols, modeling a
situation mathematically

Level III. Symbolic abstraction
Working with naked numbers, unknowns, variables, symbols

Level IV. Symbolic generalization
Identifying mathematical patterns

In looking for patterns across all participants, two general conclusions can be drawn from
the evaluation data. First and most striking is the fact that, the lower the level of
abstraction, the more successfully the participants performed. One can see this by
examining Table 6.7 below.12 To aid analysis, the scores were shaded in such a way that

the darker the shading, the more successful the participant’s score.l3 As one scans the

 

'2 A complete set of evaluation inventories—one for each of the six fields on the algebra map—can be
found in Appendix B.

'3 Entries labeled “NA” or “0” do not apply. These indicate, respectively, that a participant did not attempt
to utilize the named skill or concept, or that the interviewer did not ask/pose/present that particular question
to the participant.

I42

table from top to bottom, one notices that the lighter shades (less successful) are more
dominant lower on the chart, where the more abstract levels are shown.

Second and nearly as striking is the fact that, the stronger the participant
performed mathematically overall (across all interviews and tasks), the stronger she is at
the higher levels of abstraction. This pattern can also be seen in the shading of the table.
Participants were listed on the table from left to right in order of most successful
(Adrienne) to least successful overall (Monica) based on participants’ performance means
for evaluation items (included in Appendix B). As one scans the table from upper left to
lower right, a pattern can be seen in the table: The upper lefi half of the chart is
dominated by dark shades but gives way to light shade in the lower right half of the chart.
In other words, in Adn'enne’s and Sara’s columns (the students who performed most
successfully overall), the dark shades (indicating success) extend the farthest down into
Level IV. In Amy’s and Joy’s columns (less successful overall than Adrienne and Sara),
the dark shades begin to mix with the lighter shades (indicating some lack of success) at
Level III. In Terri’s and Monica’s columns (the students who performed least

successfully overall), the lighter shades begin to appear at Level II.

143

 

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145

Functions. The types of functions that appeared in the interview problem-solving
tasks were linear (Interview One, Task 1; Interview Three, Tasks l and 2) and non-
linear—namely, a function involving inverse variation (Interview Two, Task 3). When I
initially chose these tasks for use at the various interviews, I had not analyzed their
content to be sure that I had problems which would provide data for all areas of the
algebra terrain; so I was a bit chagrined that I ended up with what seemed like data that
was inadequate for helping me gauge what participants knew about functions from their
study of Core-Plus Course 1A.

The criteria that I used when selecting problems for the interview tasks, however,
involved finding a set of problems that was a good representation of the mathematics in
Core-Plus Course 1, Chapters 1, 2, and 3. After reconfirming for myself that the
interview tasks were, in fact, a good representation of the mathematics in these chapters, 1
was again puzzled at the small amount of evaluation items among the interview data that
pertained to functions. The next step I took was to revisit the mapping of the
mathematical content from Chapters 1, 2, and 3 (Table 6.2). What I began to realize was
that, although students are exposed to and work with various types of functions in several
places across these three chapters of the curriculum, the focus is no; on Ieaming about
concepts related to functions, their characteristics, and their behavior; rather, the focus is
on the number relationship aspects of two variables. For instance, both a “linear model
for loss” and an “exponential model for loss” are used in Lesson 2 of Chapter 2, but they
are investigated as “iterative concept[s]” using Now-Next expressions only (Coxford et
al., l998c, pp. Tl lO-TlZl). Indeed, the Teacher’s Guide points out that this was an

intentional aspect of the design of the curriculum:

146

It is important that students be encouraged to think about variables as quantities
that change — either over time or in relation to other variables. The most important
notion of variable is not simply that of a letter or an unknown. We eventually will
develop the familiar concept of function carefully. However, at this time we want
only to lay the foundation for that concept by using the phrase is a function of to
indicate a relation between variables. (Coxford et al., 1998c, p. T101; emphasis in
the original)

Nevertheless, do the data tell us anything about participants’ understanding of
functions—even if it has not been explicitly included in Chapters 1, 2, and 3? I think so.
First, students worked on three tasks involving linear fimctions. On Interview One Task
1, there were mixed results. (See first two rows of Table 6.8.) Only Sara was successful
on both items related to functions, although Adrienne, Amy, and Joy did all right on one
item or the other.'4 On Tasks l and 3 of Interview Three, however, the success is more
widespread—especially when one considers and arranges the data according to the level
of abstraction of the evaluation items. Evaluation items at the lower levels of abstraction
(other than Level IV) reveal fairly good success on the part of most participants, with
Monica the exception; although Monica, along with all but Adrienne, does show marked
improvement at Level 11 over time. Certainly, the participants possess previous

knowledge to build upon—possibly the “foundation” mentioned above?—when, in fact,

functions are introduced explicitly in the curriculum (Chapter 6 of Course 1).

 

'4 See Appendix C for work on Task 1 of Interview One by Sara, Adrienne, Amy, and Joy.

147

 

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Something similar could certainly be said for participants’ understandings of non-
linear functions as well. The participants had not encountered a fitnction of inverse
variation in their classroom studies, yet three of them (Adrienne, Sara, and Joy) had some
good, albeit incomplete, understandings of concepts related to the behavior of the

functions c = 20000/n and c = 8000/n in Task 3 of Interview Two.l5

 

'5 See Appendix C for transcript excerpts from Interview Two of Task 3 for Adrienne, Sara, and Joy.

149

 

 

 

 

 

 

 

 

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150

Number Relationshhas. The interview problem-solving tasks, unfortunately,
included only a few items that probed students’ understandings of number relationships,
all of them appearing in the problem-solving tasks of Interview Two (not coincidentally,
because Chapter 2 was the key place in the text where number relationships were a focus
of mathematical investigation). The number relationships that were explored in Interview
Two were constant and inverse variation. Upon examining the evaluation data for
participants’ performances on questions about these two types of variation, it is striking

how different the results are for constant variation and for inverse variation.

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152

However, I would venture to say that the difference relates not to the type of
number relationship at issue, but rather to the context (or lack thereof) in which the
number relationship questions were posed. In the case of Task 1 (constant variation), the
question about number relationships involves the rather abstract notion of a “Y=X line,”
whereas in Task 3 (inverse variation), the questions are closely related to the context of
the problem. This difference in participants’ performance on tasks involving
contextualized versus abstracted mathematics is among the set of categories that emerged
during the course of my data collection and ongoing analysis; it will be discussed in a
subsequent section.

Number Groups. The field of number groups, as represented on the map of the
algebra terrain, involves the study of various types of numbers—negative, real, irrational,
imaginary, and complex—and their properties. While there were a couple of interview
problem-solving tasks that made use of negative and real numbers, there were no
evaluation items that probed participants’ understandings of these number groups, nor
was there any evidence among the interview data that students possessed understandings
or awareness of number groups per se.

08

Commonalities of participants ’ understandings of algebra and functions. Having
worked my way through the algebra terrain, analyzing what participants understood about
algebra and functions, I now turn to a couple of common characteristics that I found as I
looked for patterns in participants’ understandings of algebra and functions.

Initiating and Connecting Representations. A common theme across all three sets
of interview tasks was the idea of various representations—a graph, a rule, and a table

that were different, but obviously related, representations of the same problem situation.

153

(Obvious to some, that is, but not to the participants.) This is another pattern I noticed:
that participants were not able, at this stage in their Ieaming, to move freely between and
among representations.

Interview Three Task 1 is but one of the tasks that brought this pattern to light. It
presented a table of paired data to be graphed on a scatterplot, then asked the participant
to draw a line of best fit on the graph and predict a Y-value for an X-value not found
among the data. The second part of the problem asked participants to calculate a linear
regression line for the data, and then use the same X-value, together with the equation for
the linear regression line, to predict a Y-value. In essence, the task involved using two
representations—a graph and an equation—for the same situation, making a prediction
using each of the representations, and then comparing the results. Although students
were able to create the representations and make the predictions successfully, their
understanding of the connections between the two representations seemed rather tenuous.

A second, closely related pattern I noticed was that without some support—
whether from me during the interview or from the design of the problem itself—
participants often could not initiate a representation on their own. For instance, the three
problem-solving tasks of Interview One were mathematically very similar to one another.
For each task, there were two equations to be examined for their point of intersection.
Task 2 provided a table with the rows and columns already labeled; the participant
needed only calculate the numerical amounts for the table cells. Task 3, however,
provided only a large blank space for the participant to solve the problem in any way she

saw fit.

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The results were the same across all participants: Task 2 was far easier for
participants to solve. Task 3 probably would not have been solved by any of the
participants had it not been for a great deal of assistance on my part. Without a
suggestion for how to solve the problem, the participants were at a loss as to how to
proceed. However, the more structure that was provided for solving a problem, the more
successful the participants. (Terri’s performance on Task 3 is the lone exception.)

Table 6.11

Participants’ holistic performance scores on tasks
with varying degrees of structure for problem solving

 

 

 

 

 

 

 

 

 

 

 

Interview 1 Adrn Sara Amy Joy Terri Mon
Most Structured: Task 2 4 4 4 3 3
Partly Structured: Task 1 4 4 3 l 1
Least Structured: Task 3 4 4 3 4 l

 

 

Over time, several students showed improvement in initiating representations
(Interview Two, Task 3), but seeing the connections among multiple representations
continued to be challenging for participants, especially at the conceptual (as opposed to
the procedural) level of understanding. The two examples below from Interview Three,

Task 2 illustrate this:

Adrienne did well on this problem. Her understanding of it, up through Part
E, the Table Method was great!

When we got to Part E, the Graph Method, Adrienne showed the limit of her
conceptual understanding. This was fascinating too. Adrienne got the equations
entered, and the window set, but the lines were so close together that it was
tremendously difficult to tell where they crossed (same problem I ran into when
working this problem myselfl In fact, I wish the problem had been crafied
differently so there wasn’t this problem!). So, with a little prompting from me,
Adrienne began to zoom in on the area of intersection by adjusting the window bit
by bit.

Eventually, Adrienne got to the point where she was using the Trace key to
jump back and forth between the two lines, very close to where they intersected.
She was looking at their X- and Y-values, seeing that they were very close. She

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concluded that Y=43 and X=3.5 was as close as we could approximate. Then I
asked her about what X=3.5 would mean in terms of the die. She wondered aloud
whether she should round that up to 4? Then she decided to zoom in a bit more,
and very quickly, afier zooming in for the last time, the Trace key landed her on
the point Y=48 and X=4. Still, I say that this showed the edge of her
knowledge/understanding because she was OK with the two methods (Table and
Graph) yielding different answers to this problem.

(Memo for Interview Three Task 2, 3/11/02)

Unlike any of the previous interviewees, Sara didn’t hesitate in completing any of

these problems — not even the three methods of Part E — and did them correctly,

after basic getting-started prompts from me. She wasn’t able to articulate that the
x=4 stood for the 4 that was rolled on the die, but she didn’t seem to flinch at
attempting any of the methods.

(Memo for Interview Three Task 2, 4/09/02)

While I found it natural to move back and forth between the various representations and
to select an appropriate representation to use to solve a problem task, the same was not
true for the participants. The connections between the representations seemed especially
opaque to participants, ofien occluding their use of various representations. I was
mistaken to have considered these connections to be self-evident.

Fragilitvof Understanding. According to the original design of my study, I
planned to conduct each of the three interviews with participants immediately following
the conclusion of their study in class of Chapters 1, 2, and 3, respectively. As it turned
out, however, I was unable to begin the third round of interviews until the beginning of
March, which was nearly three months after participants had concluded their study of
Chapter 3 in class. (Although the fall term extended into the second week of January,
most, if not all, of participants’ work on Chapter 3 was completed in December.) While
such a gap between participants’ work in class and their work in an interview on the same

material did not seem ideal, I realized that it would provide me with an opportunity to see

how participants’ understandings fared over time.

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To my dismay, there were two areas in which participants’ knowledge had
noticeably faded—in the use of the TI-83 graphing calculator and in their facility with
certain types of algebraic-manipulation skills. There were differences, however, in
students’ recall of what they had learned after being reminded. With the calculator, I
coached the participants along, reminding them of the steps to take to figure a linear
regression line, for instance. Although all of the participants needed help getting started
with the calculator, once they did get started, they were able to continue without much
additional assistance. The skills they had learned a couple of months back resurfaced
without all that much difficulty.

For the linear regression line calculations in Parts B and C, Amy insisted she did

not recall ever having done this in Mr. Harper’s class. I must check with him on

this! So, I led her through what to do for Part B so that we could discuss the
prediction that the calculator/data yielded and compare it to the predictions
discussed in Part A. Then, Amy was able to complete Part C pretty much on her
own. I even commented to her that she was “either a very quick study,” or that
she actually had done this in the past, because she picked up on it so quickly after

I showed her how to do the first one.

(Memo for Interview Three Task 2, 3/12/02)

Even though she didn’t recall having done the linear regression stuff in class, Sara

took right to it once I got her going. She even said “I understand now” as if, like

a toddler does, she was making it clear that she didn’t want my help because she

could do it on her own. She didn’t hesitate, when looking at the LinReg results

screen, in writing down the proper equation y = a + bx. Neither did she hesitate
in substituting the given x-value into the equation to come up with a prediction.

(Memo for Interview Three Task 2, 4/09/02)

On the other hand, some of the participants’ previous work on algebraic
manipulation did not seem to resurface, even after being shown what to do. Most striking
was their inability to apply the distributive property and to collect and combine like terms

in a problem such as Y = 2(2X) + 3(X + 4) + 4X (Task 2, Part A). Only Adrienne and

Sara were able to successfully transform the given equation to Y = 11X + 12; the others

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didn’t even know where to begin. Once the equation was in this form, however, all
participants (with the exception of Amy, whom I neglected to ask) were able to
successfully solve for X when given a Y value (with all but Joy using the technique they
were taught in class).

Although students had not made use of the TI-83 calculators in the interim
between their work on Chapter 3 and Interview 3 (according to reports from both
participants and teachers), their latent knowledge of how to use the calculator for some
nontrivial tasks returned surprisingly quickly. On the other hand, although students
probably had not made use of the their algebraic-manipulation skills in the interim
between their work on Chapter 3 and Interview 3 (during which they studied the discrete-
math-focused Chapter 4 and the geometry-focused Chapter 5), their knowledge of how to
apply the distributive property, how to collect and combine like terms—skills they very
likely had practiced in middle school math classes—did not resurface at all for most

participants.

Interpretations

The first part of this chapter has been devoted to broader data analysis: looking
for patterns in the data that shed light on characteristics of students’ previous non-
successes in school mathematics, characteristics of students’ Ieaming experiences in
Core-Plus Course 1A, and characteristics of students’ understandings of algebra and
functions. These findings, as well as the findings I presented in Chapter 5, have been
primarily descriptive. In this second part of Chapter 6, then, I offer some interpretations

of my findings.

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These interpretations share a theme. They can be used to answer the question,
What facilitated or inhibited participants’ mathematics Ieaming? Each of these
explanations emerged from the data itself. In the midst of my data collection, I began to
formulate conjectures about what I was finding. As I proceeded, I looked closely to see if
additional data would confirm and flesh out these hunches. Over time, these three 1
categories began to take shape very clearly; and, as I have proceeded with my data
analysis through writing, these interpretations have been underscored and confirmed
numerous times.
Contextualized Versus Abstracted Mghemjtifi

I have alluded several times heretofore to the differences in participants’
successes on problem-solving tasks when those tasks involve math concepts that are
presented in context versus math concepts that are presented in the abstract. The results
of participants’ successes across the different levels of abstraction—described in a
preceding section—is by far the best illustration of this finding, yet it is but one example
of many. These same differences—students’ successes with problems in context versus
students’ struggles with problems in the abstract—appeared among the types of equations
that students were able to solve as well as among the data on students’ work with number
relationships in context and in the abstract, students’ ability to use rule-generated tables
of data far better than rule-generated graphs (both on the TI-83 calculator) to find the
intersection of two functions, and students’ ability to work with scatterplots of data points
much more easily than with graphs of the functions that generated those data. It was
quite evident that, for these previously unsuccessful students, their performances on

problem-solving tasks were much stronger when the mathematics was contextualized, but

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the more removed the mathematics was from a context, the more the participants’ skills
seemed to deteriorate.

This is a fascinating finding, but a very difficult one to interpret. On first
reflection, this seems to affinn the choice of the Core-Plus authors to present
“Contemporary Mathematics in Context” (the title of the textbook series):

The curriculum builds upon the theme of mathematics as sense-making. Through

investigations of real-life contexts, students develop a rich understanding of

important mathematics that makes sense to them and which, in turn, enables them

to makes sense out of new situations and problems. (Coxford et al., 19980, p. 1,

emphasis in the original)

Participants were indeed able to make sense of important mathematics that were couched
in the contexts of real-life situations.

Yet, I can only be partly enthusiastic about this finding. First of all, I must
reserve judgment as to whether the participants were enabled to make sense out of new
situations and problems. From my study, I am only willing to agree that participants
were enabled to make sense out of new situations and problems when they, too, were
couched in a real-life context. Yet one very important and powerful use of the language
of mathematics is to express mathematical ideas in the abstract, to generalize principles
and theories that can be applied in many ways across numerous contexts (NCTM, 2000).
Given the limited data that I collected on my participants, I have not seen evidence that
they are (yet) able to engage in making sense out of new situations and problems through
the use of mathematics in the abstract. Students’ mathematics education is incomplete
without this skill.

Secondly, I have concerns about students getting to the place where they will be

able to engage in making sense out of new situations and problems through the use of

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mathematics in the abstract. I worry about Theule-Lubienski’s (1996) finding that
students ofien become stuck in the contexts of problems, losing a handle on making sense
of important mathematics couched in the contexts of real-life situations. Although
Theule-Lubienski found that the differences fell along SES lines and socioeconomic
factors were not considered in my study, I share her concern that some students are less
able to step back from the context of a problem and see the mathematical point because
of the way in which they approach contextualized problems. There were participants in
my own study who got caught up in the contexts of certain problems (Interview One,
Task 3, for instance) and consequently missed the mathematical objectives of the tasks at
times.

This leaves me with questions that my study cannot answer. Is it simply too early
in their study of the Core-Plus curriculum for participants to have reached the stage in
their mathematical development where they are facile with mathematical ideas in the
abstract? Will the participants achieve the same degree of success with abstracted
problems that they now enjoy with contextualized problems as they gain more experience
in the Core-Plus curriculum? Does the Core-Plus curriculum provide students the
learning opportunities they need to make a successful transition fi'om using mathematics
in context to making use of mathematics in the abstract? What facilitates or hinders
students’ success in making this transition? Needless to say, more research is warranted
in order to answer these questions. Moreover, it is essential to find answers, as students’
mathematics education is incomplete without the ability to use abstracted mathematics:

it sounds promising to allow students to explore math through familiar

contexts. But in Ieaming mathematics, the goal is not to stay tied to the contexts,

but to abstract the mathematical principles so they can be utilized elsewhere.
(Theule-Lubienski, 1996, p. 212)

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Language Issues

A second category that emerged during the course of my data collection and
analysis had to do with language issues—repeated occasions where participants had
difficulty with the text that accompanied a problem-solving task. It became evident that
one of the major factors that inhibited participants’ mathematics Ieaming had to do with
the language in which the problems were written. At first, I presumed that it was a matter
of the text’s being written at a reading level that was too advanced for the participants,
yet a Raygor Readability Estimate placed the reading level of the texts of the interview
tasks at about Grade 7.

What I then began to realize was that participants possessed certain limitations in
their proficiency with the more formal language that is used by the mathematics
community to convey ideas among its members (used in math textbooks and by math
teachers, for example)—i.e., the mathematics register (Pimm, 1987). Because the nature
of the interviews was interactive, it was quite easy for me, as the interviewer, to assist the
participant with the language she found difficult to navigate and to get her back on track.
Yet I wondered: In the day-to-day work of completing homework assignments or written
assessments for math class, how would a student fare if she found the text of the problem
or task difficult to interpret appropriately? I suspect that, all too ofien, what is in
actuality a lack of mathematics-language comprehension is mistaken for a lack of
mathematics-concept comprehension.

Consider the following problem posed in Interview One Task 3:

You are considering joining a karate club. At Champion’s Comer, there is no
membership fee and lessons cost $6.00 per hour. At Bowles Academy, there is a

162

yearly membership fee of $24, and lessons cost $4.50 per hour. Which club would
you join? Show or write a short explanation of how you decided.

For her answer, Adrienne chose Champion’s Comer, explaining, “If you go for only a
couple hours its [sic] cheaper.” What Adrienne wrote is correct; however, in evaluating
her response, I had to judge it as having missed part of the mathematical point of the
problem because Adrienne did not interpret the question appropriately. To one who is
acquainted with the mathematics register, it is clear that the question is asking, “Which
option is more economical?” and that the answer consists of three cases: One club is less
expensive over the short term, the other club costs less in the long run, and the key to the
answer is to find out the number of hours for which they cost the same amount. Were
this a problem on a homework assignment or a test, Adrienne would get partial credit at
best, but would not be given credit for a correct answer. Yet, during the interview, with
prompting from me, Adrienne was able to successfully arrive at the three-part answer.
Adrienne’s only error was in failing to interpret the text of the problem according to the
mathematics register.

Part of learning mathematics is Ieaming to speak like a mathematician, that is,

acquiring control over the mathematics register.

Are pupils sufficiently aware of the existence of a mathematics register which
is employed within the school?

In particular, how do pupils keep such meanings and uses separate and how do
they acquire knowledge of when to invoke one meaning rather than another? How
aware are teachers themselves when they are using various registers? Do teachers
explain these differences to their pupils, or do they assume that their pupils know
which items are occurring in the mathematics register, and when using this
particular register is appropriate? (Pimm, 1987, pp. 76-77)

It was clear from my study that participants were not sufficiently aware of the

existence of a mathematics register; in fact, I myself was only beginning to recognize my

own use of the mathematics register. Given the limited data that I collected on my

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participants and their study of the Core-Plus curriculum, I did not see evidence that such
language issues were addressed or explained to students in overt ways. At best, when
participants struggled during an interview to understand the mathematics register, I
provided a paraphrase for them, stating the problem in everyday terms. In retrospect, this
maneuver enabled participants to successfully answer the problem in question, but did
little, if anything, to help them develop their own skills in decoding the mathematics
register for themselves. Thus, although the question of how students acquire knowledge
of how to appropriately interpret the mathematical register is left to further research, it is
irrefutable that the answer to this question is the key to helping students overcome the
obstacle to success in school mathematics that such language issues pose.

“Mathematics Communigl Membership”

Issues pertaining to “mathematics community membership” are closely related to
the language issues discussed above: I came up with the term “mathematical community
membership” to refer to the collection of characteristic behaviors that mathematicians
possess, similar to the collection of characteristic behaviors of scientists that science
educators aim to instill in even the youngest of their students (National Research Council,
1996). Facility with the mathematics register is a prime example of the types of
behaviors that belong to this collection. Other examples of characteristic behaviors of
mathematicians and other members of the mathematics community include using a
systematic method when looking for all possible solutions to a problem, organizing data
into a table, recording one’s work in such a way that others can interpret one’s solution

process, using an alternate method of solution to verify an answer, and so on.

164

Figures 6.4 and 6.5 below are provided in order to give the reader an idea of the
impact on problem-solving success that even these benign mathematician behaviors can
have. Figure 6.4 is an example from the initial survey of one student’s use of a
systematic method when looking for all possible solutions to the following problem:

I have a bunch of dimes and nickels in my pocket. I need 50 cents to buy a candy
bar. List the different combinations of dimes and nickels I could use to equal 50

cents.

Figure 6.5 is an example from a student who did not utilize this mathematician behavior.

6 d’mrxes
L-l chmcs , :1 Names

37 6W0“; ‘t Nickles
.9. ders, (0 hi CKRS

1 ch me, <3 n‘. :5ch
l D m CKIQE

Figure 6.4
One student’s use of a systematic method to find all possible solutions
to the contextualized problem from the initial survey

5 DiMa3 \0 01.04459
g meta: Clad llJime

</ Dimes and c mucus

NICO Cx \O¥ MC¢

Figure 6.5
Another student’s less successful approach
to the contextualized problem from the initial survey

165

To one who already possesses membership in the mathematics community (such
as myself) by virtue of having learned these behaviors and employing them habitually,
these actions are taken for granted. Thus, it is easy to miss their significance: For those
students who do not employ the behaviors commonly used by mathematicians,
completing a mathematical task is an endeavor that is confusing and chaotic, mysterious
and discouraging. By contrast, completing a mathematical task successfully can be
facilitated by the use of these common mathematician behaviors.

Indeed, one of the major things that facilitated participants’ mathematics Ieaming
was the degree to which they had been initiated into the mathematics community—Le,
the degree to which they had experienced mathematical enculturation (Bishop, 1988).
Given the limited data that I collected on my participants and their study of the Core-Plus
curriculum, however, I did not see evidence that mathematician behaviors were addressed
or explained to students in overt ways. Unfortunately, as is the case with the
mathematics register, students are left to their own devices to learn these traits that are
essential to success in school mathematics. Further research is certainly warranted to

help develop ways in which these traits can be intentionally taught to students.

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CHAPTER 7

IMPLICATIONS

A Letter to High School Math Teachers

Dear High School Math Teachers,

As I have carried out my study of previously unsuccessful students, of their
learning experiences in Core-Plus Course 1A, and of their understandings of algebra and
functions, my thoughts have regularly turned to the question of What implications do my
findings suggest for the high school mathematics classroom? I’ve often reflected, as a
former high school math teacher myself, upon what I would do differently in my teaching
as a result of the things I have learned through this research if I were to find myself back
in the high school mathematics classroom. I would like to share my thoughts with you,
my colleagues.

(as

I have learned a lot about this whole notion of success and non-success. My own
understanding of what it means for a student to be successfirl in mathematics has changed
a great deal during the course of my research. I was guilty of simplistically categorizing
those students who received above-average grades in my classes as successful and those
students who received below-average grades in my classes as unsuccessful. I had barely
taken into consideration what students’ own experiences of the curriculum were like. I’m
afraid in doing so, I let many slip through the cracks.

I think of students like Joy or Amy, for example. Joy and Amy were practically

straight-A students and exhibited every observable characteristic of highly successful

167

students—attentive and engaged in class, asking and answering questions, completing
their assignments correctly, working productively with classmates—yet they loathed
mathematics. Because of their disdain for the subject, it’s quite probable that they won’t
choose careers related to mathematics, because they don’t see it as something relevant or
interesting. Yet, were there a way to draw them into the fold, some means of turning
them on to the excitement of studying mathematics, the possibilities are quite enticing—
to think that we’d have these young women, full of potential and the drive to succeed,
pursuing careers in mathematics and blazing a trail for other young women to follow.
Yet this will never happen if their teachers know nothing of their feelings of alienation in
school mathematics.
Schools have an obligation to ensure that all students participate in a strong
instructional program that supports their mathematics Ieaming. High expectations
can be achieved in part with instructional programs that are interesting for
students and help them see the importance and utility of continued mathematical
study for their own futures. (NCTM, 2000, p. 13, emphasis added)
I don’t claim to have an answer to the question of how to bring these young women
around. I wish I did. However, I think the place to start is by getting to know our
students and working to understand what their experiences in our classes are like.
Getting to know what my students’ experiences in mathematics are like can also
help me reach those students who are not as seemingly successful as Joy and Amy.
Who’s to say, for instance, what the reason was for Adrienne’s or Terri’s failure to pass
Core-Plus Course 1 the first time they took it?
Certain students may not be “good at math,” not because of inherent lack of
aptness in cognizing mathematical operations but because they experience as

unfamiliar or as alienating the participation structures by which mathematics is
customarily engaged in the classroom. (Erickson & Shultz, 1992, p. 477)

I68

In fact, it was true for Sara, by her own reports, that she found the participation structures
by which math was engaged in the Core-Plus classroom alienating. Could this have also
been the reason behind her failure of Eighth Grade Math as a seventh-grader? It’s
impossible for me to say. Yet I certainly believe that working to understand how students
are experiencing mathematics in the classroom is the first line of defense in the struggle
to keep them fi'om being filtered out of the mathematics pipeline (National Research
Council, 1989).

(as

I have also learned a great deal about the fragile nature of students’
understanding, both in getting it to germinate and in getting it to take root. As a result of
my research, I have developed a brand new appreciation for the difficult work that
students face in Ieaming the stuff of algebra. I’m sorry to say that I had gained too much
distance from my own first days of encountering the complex ideas of functions, multiple
representations, and various levels of abstraction. I had begun to think that those
understandings and connections, which now seem quite clear and obvious to me, were
similarly quite clear and obvious to students.

As I look back now at my data collection experiences, I’m surprised at the things
that surprised me. Why was I puzzled that Joy did not recognize the infinitely many
places where Y=X on a graph that had just two such points (Interview Two Task 1)?
Why did I marvel at the fact that Adrienne did not realize that a graph of (Atlanta
temperature, Fort Wayne temperature) did not actually help solve the problem of when
two temperatures would meet (Interview One, Task 1)? Why did I find it baffling that
Sara could not sketch the graph of Y= —2 + 3X correctly even though she could correctly

explain a negative y-intercept and a positive slope (Interview Three Task 3)? Why was I

169

so “stunned” that Amy so thoroughly confused the rules for applying the distributive
property? (Or did she?)

I was quite stunned at how little Amy knew/remembered about simplifying

equations and the rudiments of algebra. For instance, for Part A, Amy ultimately

ended up with Y = 9X ° 2X ° X + 4 and explained that she got the 9X from these

boldfaced/underlined elements in the given equation: Y = 2(2X) + ;(X+4) + 4X ,

the 2X came from the first parentheses, and the X+4 came from the second set of

parentheses. (To her credit, she wasn’t at all confident about this answer!) When

I proposed that someone else had gotten Y = 11X + 12 for an answer, Amy

admitted she had “no idea” how they got that. Even when I showed her how it

was derived, it didn’t really seem to make sense to her.

(Interview Two Task 3, 12/17/01)

If I allow myself to lose sight of just how challenging it is to wrap one’s mind
around (let alone master) such concepts, then, as a teacher, I will only be able to reach
those students for whom the Ieaming of mathematics comes most easily. I will leave
these other students behind in the dust; and if they weren’t previously unsuccessful when
they reached my class, they surely will be unsuccessful in my class. Forgetting how
complex and confusing these ideas are for the novice will prevent me from seeing the
need to offer a variety of Ieaming opportunities for students to nurture their budding
knowledge and to allow it to grow and take root.

I think that the best way to help students’ knowledge to mature is to probe their
thinking, challenge them to understand their own ways of understanding. When a student
gives a right answer, I should not make the mistake of equating that with correct
understanding; when a student gives a wrong answer, I should not make the mistake of
equating that with incorrect understanding. Only by finding out what my students think,
how they’ve reasoned through things, how they’ve interpreted the questions I’ve asked or

the problems I’ve posed, will I be able to prune the incomplete understandings and foster

new, more hardy knowledge growth.

170

(fl

Finally, I have become aware, for the first time, of how essential it is to our
students’ success that they be explicitly taught to “walk the walk” and “talk the talk” of
mathematicians. Prior to my conducting this study, I was unaware that there was a
language or a set of behaviors unique to those who have been enculturated into the
mathematics community. The need for intentionally helping socialize students according
to the ways of the mathematicians was transparent to me. Yet I have come to see how
vital this kind of instruction is for our students’ success.

Some students, like Adrienne, will pick up mathematical habits through skillful
imitation of their teachers — working methodically through a problem one step at a time,
keeping good record of their steps. Others, like Joy, will only see madness in the method
and will not follow suit — losing her way in resorting to her own devices.

Joy right away asked whether she had to do it the way she “was supposed to do it”
or the way she wanted to do it. I said she could choose. So she did this:

7k .. 3 .4 m
l0
“’5
---......
y ....- a
(Interview Two, 1/14/02)
Some students, like Joy, will seem to naturally speak the language of the mathematics

classroom with tremendous fluency. Others, like Monica, will struggle with this

language in very unexpected ways (especially if the English language is not completely

familiar):

171

I said something about “draw[ing] a line through the points” and she questioned
me about what I meant by motioning to see if I meant to connect the points dot-to-

dot. (Interview Three Task 1, 3/18/02)

If I assume that students will learn to speak the language and imitate the behaviors
that I exhibit in the mathematics classroom without my intentional help, I will be
shortchanging many students for whom deliberate instruction in these areas could be of
significant benefit to their mathematics Ieaming success. Those unable to crack the code
themselves will remain outsiders wondering at the mysterious ways of mathematics
Ieaming, possibly believing in the myth that they are simply unable to learn mathematics.

Some teachers can be reluctant to provide such unashamedly personal images [of

the way they metaphorically understand concepts], claiming that it is not at all

clear that they will be usable by anyone else, and may well confuse. However, by
not talking about such things at all, the existence of rich inner mental realms in
which mathematics properly takes place will remain undiscovered by many
pupils, who will see only the external mathematical world of symbols on paper
and operations on them. Whether or not a particular offered image is successful
in illuminating a concept, it at least serves the purpose of indicating that image-
making is an appropriate activity for pupils to be engaged in, and that the teacher

has personal images of the mathematics in question. (Pimm, 1987, p. 97)

In the same way that a master trains an apprentice, showing him how to use the
tools of the trade, how to think and act like a craftsman, I believe we, as math teachers,
ought to train our students. Through our own careful reflection on the mathematical
ways of knowing and doing that we possess, and through overt means of sharing these
with our students so that they can learn how to think and act like mathematicians, I
believe we will see greater Ieaming success among the students in our classrooms.

(93

And I trust that greater Ieaming success is the goal that you, like I, seek for your students!

Sincerely, Angie

172

A Letter to Math Teacher Educators

Dear Math Teacher Educators,

Although the research interests that motivated this study were born from my past
experiences as a high school math teacher and I have carefully considered the
implications of my findings for those who teach high school math, I have been
challenged to consider the implications of my findings for mathematics teacher educators
as well because these days I play a role in educating prospective mathematics teachers.
These are my thoughts:

As I think back on my interactions with the previously unsuccessful students who
participated in my study and on what I have come to understand about their mathematics
Ieaming experiences prior to as well as during the year that I conducted my study, it
strikes me that the mathematics reform literature addresses our obligation to effectively
reach all students without addressing the question of how to accomplish that goal.
Granted, one of the mantras of mathematics reform is “mathematics [success] for all,”
and there has been quite a bit written about how to accomplish the goal of implementing
mathematics reform. However, in my fourteen years of participating in mathematics
reform both as a math teacher and as a math teacher educator, I have not encountered
resources that address the complexities of effectively meeting the Ieaming needs of those
students who struggle.

I am thinking of students such as Adrienne and Terri who struggle once through a
course to no avail before experiencing success; students such as Sara and Amy and Joy

who struggle to find any satisfaction, meaning, or relevance in their math studies;

173

students such as Monica who struggle to understand the language of mathematics so that
the essence of what they’re supposed to be learning eludes them. These types of students
struggle with their math classes while their teachers hardly notice. (I know this is true,
based on my own experience as one such teacher.) Not only do I still puzzle over how to
effectively reach these students, I puzzle still more over how to prepare prospective
mathematics teachers to effectively reach these students.

(as

One insight I have garnered from my research is that teachers must make it their
business to find out what their students’ mathematics Ieaming experiences are like from
those students ’ perspectives. This is a particular challenge when I realize that, based on
my own school math experiences, my perspective of mathematics Ieaming is quite
different from that of students like my study participants; somehow I just succeeded in
math with very little effort, always enjoyed my math studies, and ...developed fairly
well-rooted conceptual understandings of the mathematics on my own. Yet this cannot
be expected for students, like my study participants, for whom math achievement does
not come easily, for whom the “real-life” contexts of math problems are not meaningful,
or for whom the mathematics register is not a familiar language.

Another insight I have garnered from my research is that teachers must accept it
as their responsibility to intentionally socialize their students into mathematical ways of
thinking, speaking, and behaving. This, too, is a particular challenge when I realize that,
in my own school math experiences, I was never explicitly socialized to become a
mathematics community member; somehow I just acquired a large repertoire of
mathematician behaviors on my own. Yet it is not enough to hope that this will just

happen for students who, like my study participants, could be that much more successful

I74

if they were explicitly taught mathematical modes of thinking, speaking, and behaving.
Just as Lisa Delpit (1988) maintains that the rules of “the culture of power” must be
explicitly taught to children who grow up outside of it, as a result of this study, I maintain
that the same is true about the culture of mathematical power.

05

So how do we, as mathematics teacher educators, teach prospective (and
practicing) math teachers to attend to their future students’ Ieaming experiences? How do
we, as mathematics teacher educators, teach teachers to attend to their future students’
mathematical enculturation? Both of these are questions to which I do not yet have
complete answers. Nevertheless, it is imperative that expertise in each of these areas be
acknowledged as essential components of pedagogical content knowledge of
mathematics.

I believe that the place to start is by attending to our own students’ Ieaming
experiences in, and of, the math-teacher education classroom. As a math teacher
educator, I must model for and with prospective math teachers the process of attending to
students’ (i.e., again, their own) Ieaming experiences and to students’ (i.e., their own)
mathematical enculturation. Math teacher educators can learn by doing; similarly,
prospective-math-teacher students can learn what it is like to learn by doing as they see
their instructor Ieaming by doing. This work is a bit like navigating a hall of mirrors,
which is admittedly tricky, but important nonetheless (Lampert & Eshelman, 1995).
Without creating ways for prospective teachers to learn to understand their students’
math-Ieaming experiences and to intentionally socialize their students into mathematical
ways of thinking and behaving, I believe many more school math students will fall

through the cracks of mathematics reform.

175

(93
And I trust that finding answers to the question of how to keep school math students from
falling through the cracks is the goal that you, like I, seek for your prospective-math-
teacher students!

Sincerely, Angie

176

APPENDICES

177

APPENDIX A

DATA COLLECTION INSTRUMENTS

178

 

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179

Initial Survey Instrument

Please respond to the 18 statements below. For each one, mark an X in one of the boxes to show
whether you Strongly Agree, Agree, have No Opinion (or are Undecided), Disagree. or Strongly
Disagree with the statement. Please complete the two math problems on the back as well.

Thank you very much for your time!

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Ajfiuons

 

1.

I often volunteer to answer questions in math class.

 

2.

3.

I understand math well enough to help a classmate who is
stru lin .

 

I do n_qt like to go to the board to show my answer to a math
homework problem.

 

:5

I am able to tell for myself whether an answer on my math
homework is correct (or at least reasonable) without looking
up the answer.

 

Math often does ngt make sense to me.

 

I am naturally good at math.

 

Math was one of my favorite subjects in middle school.

 

@NP’S"

I love playing math games in class where you have to be the

fastest one to answer in order to win.

 

S0

I can imagine myself completing a math-related major in
college.

 

10.

I want to get a job that uses as little math as possible.

 

12.

11. A student doesn’t need more than 2 years of math in high

school if he (or she) is not going to go to college.

 

A student doesn’t really need to take 4 years of math in high
school unless she (or he) is going to be a math major in

college.

 

13. I would be happy to get a C as my grade in Integrated Math

1-2.

 

14. I do _n_ot plan to take more math courses beyond the two that

are required to graduate from high school (Integrated Math
1-2 and lntegated Math 3-4).

 

15

. I want to graduate from high school with an Academic
Honors Diploma.

 

16

. When a math problem is challenging, I usually keep working

at it until I get it figured out.

 

17

. I usually check over my answers on a math test before
tuming it in.

 

18

 

. When I get back my graded math tests, I usually go over the

problems I missed so I understand why I got them wrong.

 

 

 

 

 

 

180

 

Initial Survey Instrument Scoring Rubrics (cont’d)

1. I have a bunch of dimes and nickels in my pocket.
I need 50 cents to buy a candy bar.
List the different combinations of dimes and nickels I could use to equal 50 cents:

2. Complete the table using the equation 5y = 50 — 10x

 

 

 

 

 

 

 

 

181

Interview Protocols

These are sample questions for the individual interviews with students. Specific questions will depend

upon each student’s responses to agree/disagree statements and problem-solving tasks that have been
completed.

PROTOCOL A1:
Questions about the student’s personal math history

0 Tell me your opinion of math as a school subject.
What do you enjoy most about math?
What do you enjoy least about it?
How well would you say you do in math?

0 Describe to me one of your most successful experiences in a math class.
0 Describe to me one of your least successful experiences in a math class.

0 Do your parents ever help you with your math homework?
If yes: Give me an example.
If no: Why not?

0 Do you ever use math outside of school?
If yes: Give me an example.
If no: Why not?

PROTOCOL A2:
Questions about the student’s recent experiences in the Integrated Math course

0 Tell me your opinion of Integrated Math 1-2 as a class.
What do you enjoy most about the class?
What do you enjoy least about it?
How well would you say you are doing in the class?

0 What did you study in class this past week?

0 Did you participate in the class discussion this past week?
If no: Why not?
If yes: Give me an example.
Did your answer happen to be wrong?
If yes: Give me an example.
How did you know your answer was wrong?
How did you feel?
How did the teacher respond?

0 Did you contribute any ideas to a small-group investigation this past week?
If no: Why not?
If yes: Give me an example.
Did anyone disagree with your idea(s)?
If yes: Give me an example.
How did you know that someone disagreed with your idea(s)?
How did you feel?
How did your classmate(s) respond?

182

Interview Protocols (cont’d)

PROTOCOL B:

Questions about the student’s understandings of the problem-solving task(s)

How confident are you that your answer to this problem is correct?
Tell me why you feel this way.

Explain to me what you were thinking when you solved this problem.

Is there any other way to solve that problem?
If yes: Explain it to me.

Is there any other answer that would (also) be correct?
If yes: Explain it to me.

In the event that the student '3 answer is (partially) incorrect:

I know of someone else who worked on this problem. She came up with (correct answer) for an
answer. How is that answer different from yours?

Can you imagine how she might have gotten this answer instead of the one you got?

If yes: Explain your thinking to me.

 

Would you like to revise your answer?
If yes: Explain your revised answer to me.

Imagine that you are showing this problem to an 8m-grade friend who is not taking Integrated Math
right now. Tell me what you would say to her/him so that s/he understands what the problem is
asking.

How would you explain your method of solving the problem so that your friend is convinced that your
answer is correct?

183

Interview Attitude Questionnaire

Please respond to the 10 statements below about your experiences in your Core-Plus
Course 1 math class.

For each one, mark an X in one of the boxes to show whether you Strongly Agree,
Agree, have No Opinion (or are Undecided), Disagree, or Strongly Disagree with the
statement.

(I)

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m ‘1

 

1. I take part in whole-class discussions on a regular
basis.

 

2. I contribute my fair share of ideas during small-group
time.

 

3. Most of the things we're Ieaming in Core-Plus
Course 1 make sense to me.

 

4. If I work at it hard enough, I know I can succeed in
this class.

 

5. I am interested in the things we’re Ieaming in Core-
Plus Course 1.

 

6. Based on my experience in Core-Plus Course 1, I am
looking forward to taking Core-Plus Course 2.

 

7. The problems we work on help me see the
usefulness of math in everyday life.

 

8. I find it helpful that we use graphing calculators in
this class.

 

9. When a math problem is challenging, I usually keep
working at it until I get it figred out.

 

10. I am putting forth my best effort in my Core-Plus
Course 1 class.

 

 

 

 

 

 

 

 

184

Interview One Familiarity Checklist

Below is a list of the topics in Unit 1 of your math book.

For each one, mark an X in one of the boxes to show which statement best describes
how familiar you were with the topic before m semester.

 

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waqt inoqe em" Men meux | 1nq
‘eJOIOQ eseq; uees JeAeu pen |

new Men esem op 01 moq meux ApeeJle |
main tnoqe Bungue M0u>1 iou pgp | pue

'1equuJos eseut op 01 mou M9U)| Apeeije I
'I nun ug eseqt Bug/Ipnts Jeqwewai Luop |

 

Stem-and-leaf plots

 

Number line plots

 

Histograms

 

Mean, median, mode

 

Five-number summary

 

Box plots

 

Scatterplots

 

Plots over time

 

 

 

 

 

 

 

 

185

Interview One Task 1

1. At 3:00 P.M., the temperature is 86° F in Atlanta and is
decreasing at a rate of 3 degrees per hour.

At the same time, the temperature is 56° F in Fort Wayne and is
increasing at a rate of 2 degrees per hour.

When will the temperatures be the same?

Show how to use a graph to solve the problem:

 

186

Interview One Task 2

2. You want to join a DVD club to buy movies on DVD.
Your friend says that it is more economical to buy the DVDs at the mall.

You decide to consider your options:
The DVD club has a membership fee of $40 and each movie costs $20.
The store in the mall charges $30 for each DVD.

a) Complete the table.

 

Number of DVDs 1 2 3 4 5 6 7 8

 

DVD Club Cost (8)

 

Mall Cost (S)

 

 

 

 

 

 

 

 

 

 

 

b) What would you say to your friend about which option is more economical?

187

Interview One Task 3

3. You are considering joining a karate club:

At Champion’s Comer, there is no membership fee and lessons cost $6.00 per hour.
At Bowles Academy, there is a yearly membership fee of $24, and lessons cost $4.50 per
houn

Which club would you join?

 

Show or write a short explanation of how you decided:

188

Interview Two Familiarity Checklist

Below is a list of the topics in Unit 2 of your math book.

For each one, mark an X in one of the boxes to show which statement best describes
how familiar you were with the topic before fills semester.

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‘aJOjeq asaqt uaes peu |
‘eJOqu esaqt uees JeAeu peu I

"am Men 33an op 01 meg meux Apeeije |
'teumeuios aseqt op 01 moq maux ApBBJle l
'weqi tnoqe Bugqtliuemoux iou mp | pue

'z tgun u! SILII fiuMpnts Jequieuiei ;,uop l

 

TOPIC
_—
Using a y = x line

 

Graphing (x, y) data

 

Writing a NOW-NEXT expression

 

I Using the ANS key to calculate numbers
repeatedly

 

Writing a rule using letters

 

I Using a rule to produce a table

 

I Choosing Xmin, Xmax, Ymin, Ymax to set
the viewing window

 

I Using a rule to produce a graph

 

Using rules that produce non-linear
graphs

 

 

 

 

 

 

 

 

189

Interview Two Task 1

1. Some students are better at some school subjects than they are at others. The end-
of-semester English and mathematics grades of 12 students are given below.

English 95 73 84 90 7O 64 80 74 90 88 80 76

 

Mathematics 85 75 80 66 87 83 91 74 94 74 80 63

a) Make a scatterplot of these grades on the axes below. Draw the line y = x on your
scatterplot.

a, 100
o
‘8
5 90
(D
.g 80
E
o 70
5
m
E 60

 

60 70 80 90 100

English Grades

b) How many students had a higher grade in mathematics than in English?

c) How many students had a higher grade in English than in mathematics?

d) Some people say that mathematics is more difficult than English.
Do these data support that statement?

190

. Interview Two Task 2

2. Sara has a weekend job as a waitress at her family’s restaurant. She works 8 hours
on Saturdays and 4 hours on Sundays. Her pay is $2.50 per hour plus tips.

a)

b)

d)

6)

Choose letters to represent the variables daily income and daily tips. Write a rule
that gives Sara’s income on Saturday as a function of her tips for that day.

Letter for daily income Letter for daily tips Rule -

 

Write a rule that gives Sara’s income on Sunday as a function of her tips for that
day.

Rule

 

One Saturday, Sara earned $70.00 in tips. Use your rule to calculate her income
for that day.

Saturday income

 

The next day was Mother’s Day, which is usually a good day for tips. If Sara
earned $80.00 in tips on Mother’s Day, how much did she earn that day?

Mother’s Day income

 

How does the amount that Sara earned on Sunday (in part d) compare to the
amount she earned on Saturday in part c?

191

Interview Two Task 3

3. Oliver High School’s seniors would like to take a class trip to Washington, DC. The
tour company offers to arrange the entire trip for a set fee of $8,000. This cost is to
be shared annually by the seniors who go on the trip. If n seniors go on the trip, you

8000

n

 

can calculate the cost 0 to each one by the rule 0 =

a) Complete the table below. Show (or .3 explain) your work.

Number (n) 20 40 60 80 100
Cost per Person (c)

Show work:

b) On ja sepirge sheet of paper, make a graph of the (number, cost) relation.

‘6 Explain to me, as you make your graph, what you are doing and what you are
thinking.

c) As accurately as possible, complete this sentence describing the pattern relating
the number of seniors going on the trip and the cost per person.

As the number of seniors going on the trip increases

‘6 Explain how the shape of the graph for part b matches the description you
wrote above.

192

d)

Interview Two Task 3 (cont’d)

Some seniors were trying to get the others to agree to a senior trip to Hawaii
instead of to Washington, DC. The tour company could arrange that, too, but the
total cost would be $20,000. Write a rule that gives the cost 0 per senior going
on a trip to Hawaii for various numbers n who go.

Rule

 

.5 Predict how the shape of the graph for the Hawaii trip will look compared to
that for the Washington, DC trip.

On the graph th_a_t you made for part b, make a graph of the (number, cost)
relation for the Hawaii trip.

.3 Explain to me, as you work on the problem, what you are doing and what you
are thinking.

.6 Explain how the cost for the Hawaii trip compares to the cost of the trip to
Washington, DC.

193

Interview Two Task 3 (cont’d)

Problem Task 3, Parts B & E

 

194

Interview Three Familiarity Checklist

Below is a list of the topics in Unit 3 of your math book.

For each one, mark an X in one of the boxes to show which statement best describes
how familiar you were with the topic beIQLe ygu studied Unit 3 this school year.

 

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‘qumewos eseui op 01 mou M8U>| ADBBJ|B l
'weqi inoqe emu lIJeA meux l 1nq

‘eJOjeq eseqi uees peu |

main inoqe Bugqulue moux iou mp | pue
‘eJOjeq eseut uees Jeneu peq |

'9 nun u! Slut bug/Ipms JOQUIOIUOJ 1,uop |

TOPIC

 

Draw a line to fit the pattern in a plot

 

Make predictions using a linear model

 

Finding the rate of change, the slope, and
the y-intercept of a linear graph

 

I Find the linear regression model for a set
of data

 

Use the equation of a linear model to
make a quick sketch of the graph by hand

 

I Using a table to find values of variables
that satisfy the conditions of two
equations or inequalities

 

Solve an equation such as 3x + 12 = 45
without the use of a table or graph

 

Writing equivalent equations by
rearranging, combining, and expanding
terms

 

 

 

 

 

 

 

 

195

Interview Three Task 1

1. The following table gives heights of eight tall buildings in the United States in stories,

in feet, and in meters.

Atlanta, _, ,

. Plaza
First

World Center

 

77

73

" ”Aima'n'aei‘ lidsion:

Below are two scatterplots, one of (number of stories, height in meters) and the other

of (height in feet, height in meters).

a) For each scatterplot, draw a linear model that you believe is a good fit for the

trend in the data.

.6 Explain why you drew the lines where you did.

i 450

.5
8

Height in Motors
0)
8
e
e

‘2’

30 50 70 90 110
Mm ber of Stories

For the scatterplot on the left, use
your linear model to predict the
height in meters of a 90-story
building:

 

196

 

 

500 #77
450 .
400 .

35o. . .'.

Height In Meters

_ e‘ ‘
300 .': - l - ~ 747 A.
250-, l ., , . . .

1000 1100 1200 1300 1400 1500

Height in Feet

 

For the scatterplot on the right,
use your linear model to predict
the height in meters of a 1400-
foot building:

 

 

b)

C)

d)

Interview Three Task 1 (cont’d)

Write an equation for the linear regression line for the (number of stories, height
in meters) data. Use it to predict the height in meters of a 90-story building.

Linear regression line: y =

 

Predicted height of 90-story building:

 

Write an equation for the linear regression line for the (height in feet, height in
meters) data. Use it to predict the height in meters of a 1400-foot building.

Linear regression line: y =

 

Predicted height of 1400-foot building:

 

Compare the two scatterplots in part a. Based on your comparison, in which
estimate would you have more confidence, the one in part b or the one in part c?
Explain why.

More confidence in part b part c

Explanation:

197

2. Jerome and Lin invented a game. They take
turns throwing 10 darts (one dart at a time, of
course) at a target like the one shown at the
right. After each has a turn, one rolls a regular
die (with 1 to 6 dots on each face). The result
of the roll of the die is x, which they then use
with the equations corresponding to where their
darts landed to compute the number of points
each player earned for that turn.

a)

b)

Interview Three Task 2

 

In his first try, Jerome had two darts in the

bull’s-eye, worth 2x points each; three in the

second ring, worth x + 4 points each; and

four in the outer ring, worth x points. One

dart was outside the outer ring. On his first

turn then, Jerome earned y points where y =

2(2X) + 3(X + 4) + 4x.

Write an equivalent equation in the form y = a + bx. Show how you got your
answer.

Equivalent equation: y =

 

After rolling the die, Jerome found that he had earned 45 points. Write and solve
an equation whose solution is the number that Jerome rolled on the die.

Equation:

 

Solution: x =

On her first turn, Lin had one dart in the inner bull’s-eye, four in the second ring,
and four in the outer ring. Write an equation that gives y, the number of points
that Lin made on this turn, in terms of x.

y:

 

198

d)

9)

Interview Three Task 2 (cont’d)
Use the results of parts b and c to find the number of points Lin earned on her
first turn. (Use the same number Jerome rolled for x.) Show and explain your
work.
Lin ’3 points:

Work and explanation:

On their second turns, the equation for Jerome’s points was equivalent to y = 8 +
10x, and the equation for Lin’s points was equivalent to y = 12 + 9x. Before
rolling the die to get x, they wondered what value of x would give them the same
number of points. Find the answer to their question, and show how it can be
found using a table, a graph, and an equation.

Value of x that gives Jerome and Lin the same number of points: x =

How to use a table:

How to use a graph:

How to write and solve an equation:

199

Interview Three Task 3

3. Three lines I1, ,2, and I3 are graphed below. As you Ieamed in Unit 3, each line has an
equation of the form y = c + dx.

 

 

 

Y A I;
’3
‘ >
X
\.
V

a) Describe c and d for each of the three lines as completely as you can from the
information given in the graph. For example, indicate which values are positive,
negative, or zero, and why you think so.

Description for line [1:

Description for line [2:

Description for line l3:

b) Are the values of c equal in the equations of any two of the three lines l1, l2, and
I3? Are the values of d equal? Explain your answer.

Values of c: Yes No
Values of d: Yes No
Explanation:

200

APPENDIX B

INSTRUMENTS OF ANALYSIS, DATA SUMMARIES, AND
LONGER DATA EXAMPLES

201

Initial Survey Instrument Scoring Rubrics

Please respond to the 18 statements below. For each one, mark an X in one of the boxes to show
whether you Strongly Agree, Agree, have No Opinion (or are Undecided), Disagree, or Strongly
Disagree with the statement. Please complete the two math problems on the back as well.

Thank you very much for your time!

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

problems I missed so I understand why Lgot them wrong

 

 

 

 

 

<12 3
g > c g 9. 9. (,2
3 <2 39 8 8’ 8
> g 8 5' ‘9‘ ‘9‘ t8
‘9. a 53>; i i ‘2
8 e
1. I often volunteer to answer questions in math class. 5 4 3 2 1
2. I understand math well enough to help a classmate who is 5 4 3 2 1
struggling.
3. I do ggt like to go to the board to show my answer to a math 1 2 3 4 5
homework problem.
4. I am able to tell for myself whether an answer on my math 5 4 3 2 1
homework is correct (or at least reasonable) without looking
up the answer.
5. Math often does mt make sense to me. 1 2 3 4 5
6. I am naturally good at math. 5 4 3 2 1
7. Math was one of my favorite subjects in middle school. 5 4 3 2 1
8. I love playing math games in class where you have to be the 5 4 3 2 1
fastest one to answer in order to win.
9. I can imagine myself completing a math-related major in 5 4 3 2 1
collegg
10. I want to get a job that uses as little math as possible. 1 2 3 4 5
11. A student doesn't need more than 2 years of math in high 1 2 3 4 5
school if he (or she) is not going to go to college.
12. A student doesn’t really need to take 4 years of math in high 1 2 3 4 5
school unless she (or he) is going to be a math major in
college.
13. I would be happy to get a C as my grade in Integrated Math 1 2 3 4 5
1-2.
14. I do go_t plan to take more math courses beyond the two that 1 2 3 4 5
are required to graduate from high school (Integrated Math
1-2 and Intefiated Math 3-4).
15. I want to graduate from high school with an Academic 5 4 3 2 1
Honors Diploma.
16. When a math problem is challenging, I usually keep working 5 4 3 2 1
at it until I get it figured out.
17. I usually check over my answers on a math test before 5 4 3 2 1
turning it in.
18. When I get back my graded math tests, I usually go over the 5 4 3 2 1

 

202

 

1.

Initial Survey Instrument Scoring Rubrics (cont’d)

l have a bunch of dimes and nickels in my pocket.
I need 50 cents to buy a candy bar.
List the different combinations of dimes and nickels I could use to equal 50 cents:

SOLUTION:

 

Dimes 0 1 2 3 4 5

 

 

 

 

 

 

 

 

 

 

Nickels 10 8 6 4 2 0
SCORING RUBRIC:
0 no attempt
1 one correct pair
1.5 two correct pairs
2 six correct (listed values/not it of d,n)
or three incorrect/unreasonable
2.5 three correct pairs
3 four correct pairs
3.5 five correct pairs
4 six correct pairs

Complete the table using the equation 5y = 50 — 10x
SOLUTION:

 

 

 

 

 

 

 

 

SCORING RUBRIC:

0 no attempt

1 major errors

2 decreasing values, proportionally-spaced; no work shown
3 all but last step OK; nearly correct work shown

4 correct, or correct with very minor error

203

Table 3.1
Initial Survey Data Summary

 

  

 

  

 

  

 

 
  
    

 

 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I- III In In In
9 Z-Score for Context Problem "I ". F. '7 F.
a G G 6 Q G
I-
; ScoreonContextualProblem mgvomg v v 2v 2 v v v— vmm
>
'2 Z-Sc r Ab P l 3 S. 3 '5 9
3 ore or stract rob em 9. c. 9. § _1
.n
2
9- ScoreonAbstractProblem O~°°¢~°~°°~MON
Z—Score for Short-range Stick-t 3. S: ‘3: g 8
it ? ? ? 9 O
Stick-to-it-lve-ness, Short-range 2 f: 3 2 :- 3 S; 2 2 :3, 3 - :3 3 3 3 2 3,:
Z-Score for Long-range Stick-t a. g g 23 33
it ? G v- ? ?
Stick-to-it-ive-ness, Long-range I: ,l: E: 2 : ° 2; :3 2 3 2 ° ,3: - ‘3 Z a 2
z- r i :2 3‘- 7) 5 :2
Score or S gnitlcance ?' c. e. c. ?'
8
GI oonnovonmco b o no
g Significance éédcfiéfihiNNrécdac-dzid'vid
E :2 z :2 3 e
Z-Score for Enthusiasm - - - - "i
g" e a e e e
U
OOOOOI‘OMWGOI‘OOBFI‘B
Enthusiasm é—Jdérfiééééfidédfidfidé
'3 s 8 3 '“
O O c
Z-Score for Self-Cont, Personal ,7 a“ ? ? o‘
Self-confidence, Personal 3 2 :3 2 S; : f: 2 :3 ' :1 tr: 2 ' 2 :3 S. 2
3 I! I! ° 3
O O I “
Z-Score for Self-Cont“, Public ? ? ? _: e‘
Selreonndencemuc SMSSS-SZSSSeES-iSES}
o—ON'OBOWWNMO‘O‘v—"mmrflm
z. noaqthhhmqqqqaaafi
ScoreforResponseMean wa'j-Q???C,>9O,<?C,>C,>oooo
RankingofResvonseMean idddooooooooooBEBB
2222
ResponseMean 88$8§8$§3833883E22
——v—NNNNNNNNNMMMMMM
E I g
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7MMNNNNNNNNNN—H————

 

204

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3:: Ed :3 33 Nde 39° ammo $3 $3 madam
NN.N a: 92 EN NN.N _ _.N NN.N 5N NN.N ”“502
8o :82

8: mad 3: 22 NN_._ 2: $3 $8 id “3on
8N one 9% 8N NN.N 5N EN 2a 8a as»:
3 go so:

3.... on: $2 E: B..— 83 N2... NNE SS .355
m...” E... _— n...n RN 23 SN N.N..N :3 8d "882
—n ho flue—Z

v o 3 3. o.“ 3. 3 3 SN 32 N? _

v v 3 3 3 N... 2. 2. a: 5 a? N

v o S. 3 3. 3 N.N 3. a: 5 ~ 3 m
on _ 3 3 3 3 E N.N 2: 5 NN.N v
v N 3. on 3 N.N 3. N.N 3o 5 NN.N n

v N N.N 3 3 E N.N 3. 3° 5 com o

N o N.N QM N.N N.N S. N.N ”no 5 31.. N
2 o N.N 3 N.N N.N 2. 3 one 5 NN.N N
c o «N M: E M: 3 3 one 5 NN.N 3
a... v NN.N v 8... a... 3... 3. n3. N.N 9N... N.N an... 3. 2.. 3. e3 5 NN.N cum ..3
nN o N.N qN N.N N.N N.N N.N «N5 5 a; :
«N c 3. ca 3 N.N N.N oN _N.o 5 ta ..2
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205

Table 3.2
Explanation of Evaluation Item Types

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

U Comprehension of what the problem was asking;
Understanding the nature of the problem
E Extension; Going beyond what the problem is asking to explore ideas
about the mathematics
T Skills that support using a table for solving
G Skills that support using a graph for solving
R Skills that support using a rule (equation) for solving
P Skills that support using the coordinate plane for solving
M Skills that support using symbolic manipulation for solving
J Conjecturing; Using what you already know to form a conjecture
TS Items where the table aided the student in obtaining the solution
GS Items where the graph aided the student in obtaining the solution
RS Items where the rule (equation) aided the student in obtaining the
solution
PS Items where the coordinate plane aided the student in obtaining the
solution
MS Items where symbolic manipulation aided the student in obtaining the
solution
J S Items where the conjecture aided the student in obtaining the solution
C Calculator skill

 

 

206

 

Table B.3

Interview One Tasks: Listing of Evaluation Items and Item Types

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2:1:(ew $3; valuation Item (skill or concept)

Intl - Tl U Vocabulary - "At a rate of .Jer ..."

Intl - Tl U Understanding that the question implies At what time are the temm‘atures the some?
Intl - Tl TS Using paired data to find a point of intersection of two functions

lntl - Tl GS Uflggraph to find a point of intersection of two functions

Intl - Tl G Graphing Time versus Temperature

Intl - T] G Choosing appropriate ranges for axes

Intl - Tl G Choosing appropriate scales for axes

Intl - Tl G nterpolating for points that do not fall on the grid lines

Intl - T1 G Understandingwhere the origin is on the coordinate plane

Intl - T2 T Correctly completing a dual table of data

Intl - T2 U Recognizing that the answer has more than one case

lntl - T3 U Understanding that the question implies Which option is more economical?

Intl - T3 U Recogniizing that the answer to the question has more than one case

Intl - T3 TS Using paired data to compare two functions

lntl - T3 RS Usig an explicit formula to compare two functions

Intl - T3 R Deriving NOW-NEXT equations for calculating values

Intl - T“ C Using the Answer Key on the TI-83 to quickly calculate values iteratively (Any Task)

 

207

 

Table B.4

Interview Two Tasks: Listing of Evaluation Items and Item Types

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

allelrview & #5:; [Evaluation Item (skill or concept)
Int2 - Tla G Interpolating for points that do not fall on the grid lines
lnt2 - Tl a G Drawing the y=x line accurately on Maph
Int2 - Tl a U Understanding what the y=x line represents

Recognizing that there are many places on graph (other than data points) where
Int2 - Tla 'U y=x
Int2 - le GS Usingpoints where y>x to answer something about the overall data
Int2 - le TS Identifying points on the TABLE where y>x (or y<x or y=x)
Int2 - Tlc GS Using points where y<x to answer something about the overall data
Int2 - Tlc GS USE points where y=x to answer something about the overall data
lnt2 - Tld E Knowing what the data do/not suggest
Int2 - T2a/b R Translating words into an algebraic representation/expression
Int2 - T2a R Assigning letters to variables
Int2 - T23 R Writing equivalent expressions/equations
Int2 - T2e/d RS Usirg the equations Qules) to find total earnings

Being able to explain why two equation-values are the same if their "constants"
Int2 - T2e U differ
Int2 - T3a RS Usig a rule to find the y-valucs for a table of data [Washington DC trip]
Int2 - T3b G Using appropriate axes (x-axis for independent variable)
Int2 - T3b G Choosing appropriate ranges for axes
Int2 - T3b G Choosing appropriate scales for axes
Int2 - T3b G Interpolating for points that do not fall on the grid lines
Int2 - T3b C Entered one set of data into lists to make scatterplot representation
Int2 - T3e U Beingable to describe the relationship between the function variables
Int2 - T3d R Writing a rule (formula) fomiven situation [Hawaii trip]
Int2 - T3d RS Used Washington rule as a model for Hawaii rule
Int2 - T3d J Predicting the shape of the graph for a function similar to one already graphed
Int2 - T3e (fxn) C Graphed one function on calculator by entering rule as Y=

Graphed two functions on paper (using same set of axes appropriately)
lnt2 - TBefon) GS Complementary Strategy

Entered two rules as Y: and examined table values for comparison
Int2 - T3e (fer TS Complementary Strategy
Int2 - T3e (fxn) C Entered two seg of data into lists to make dual function representation
Int2 - T3e (fxn) C Graphed two functions on calculator by entering rules as Y=
Int2 - T3e T Made her own written table
Int2 - T3e J Predictingp y-value for an x-value not in the data set
Int2 - T3e E Understanding the infinite extent of a function

Making a reasonable conjecture about the two numerical patterns as the functions
Int2 - T3e J extend toward x-approaclfig-positivc-infinity
Int2 - T3e JS Using the '1‘1-83 to check out conjectures
lnt2 — T3f U Being able to describe how the two functions compare
Int2 - T 3e ifxn) C Graphed one function on calculator bLenterig rule as Y=

 

208

 

Table B.5

Interview Three Tasks: Listing of Evaluation Items and Item Types

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

[113:3 iew & 43;; [Evaluation Item (skill or concept)

Finding a linear model that is a good fit for the trend in the data by eyeing with a
Int3 - Tla G clear ruler

Understanding which line is the best fit and knowing why (when eyeing with a
lnt3 - Tla U clear ruler)

Using the estimated linear model to predict a y-value for an x-value not in the
Int3 - Tla GS data set

Understanding the connection between the scatterplot representation and the
lnt3 - Tla U context situation

Understanding what makes a good estimate and why (cf. interpolating betw 80-
lnt3 - Tla E nd llO-story bldgs)
lnt3 - le/c C Using the Tl-83 to find the Linear Regression line (y = a + bx)

Using the linear regression model to predict a y-value for an x-value not in the
Int3 - le/c RS data set

Understanding of what makes a good estimator and why (lines where r=.87 and
Int3 - Tld E r=.999998)
Int3 - T2a M Manipulating expressions/equations -- applying the distributive property
lnt3 - T2a M Manipulating expressions/equations -- collecting & combining like terms
lnt3 - T2b M Using a given y-value to solve for X --> y = 11x + 12 [Type Beta]

Translating words into an algebraic representation/expression (using a similar,
lnt3 - T2c R worked-out example)
lnt3 - T2d RS UsinLagiven x-value to solve for Y --> y = 1(2x) + 4(x+4)+ 4x [Type Alpha
lnt3 - T2e] (fxn) T Entered two rules as Y= and examined table values for comparison
Int3 - T2e] (fxn) TS Finding a point of intersection of two functions (TI-83 table)
Int3 - T2e] U Understanding what the x- and y-values represent from the problem context
Int3 - T2e2 (fxn) C Graphed two functions on calculator by entering rules as Y=
Int3 - T2e2 GS Finding a point of intersection of two functions (TI-83 graph)
Int3 - T2e2 U Understandingwhat the x- and y-values represent from the problem context
lnt3 - T2e3 MS olving for X --> 8 + 10x = 12 + 9x [Type Gamma]
lnt3 - T3a U Knowing that "y-intercept" describes the place where a line crosses the y-axis

Identifying positive and negative y-intercepts, given 3 lines drawn on the
Int3 - T3a P coord plane
lnt3 - T3a U lKnowing that "slope" describes the uphill or downhill characteristic of a line
Identifying positive, negative, and zero slopes, given 3 lines drawn on the

Int3 - T3a P coord plane
Int3 - T3b PS Correctly identifyirg lines with saw-intercept
lnt3 - T3b PS Correctly identifying lines with same slope

Using knowledge about the y-intercept value appearing in y = -2 + 3x to plot a
Int3 - T3e U oint at (0,-2)

Using knowledge about the slope value appearing in y = -2 + 3x to sketch an
Int3 - T3c U "uphill" line thro y-int

Understanding what C and D represent in an abstracted linear equation --> y =
lnt3 - T3a U + Dx
nt3 - T3a U owingwhich of C or D is slope and which is y-intercept

 

209

 

=a .2

 

FREE—5 28m :32 nouns—9H :25.
9mm 03a?

2l0

List of Objectives

I. Unit 1 Objectives:

C. To compare sets of data using scatterplots and the line y=x, and to interpret these comparisons for
the real-world contexts that gave rise to the data

4.

Lesson 4 Objectives:

a. To use scatterplots to find an association between two variables
b. To use the line y=x to analyze data on a scatterplot

c. To compare the information provided by various types of plots
(1. To analyze plots over time

II. Unit 2 Objectives:

A. To begin developing students’ sensitivity to the rich variety of situations in which quantities vary
in relation to each other
To develop students’ ability to represent relations among variables in several ways — using tables
of numerical data, coordinate graphs, symbolic rules, and verbal descriptions — and to interpret
data presented in any one of those forms
To develop students’ ability to recognize important patterns of change in single variables and
related variables

C.

l.

2.

3.

4.

Lesson 1 Objectives:
a. To identify key variables in the situation to be modeled

b. To collect data that will suggest the pattern relating those variables
c. To make a table and graph the data to look for patterns
(I. To make predictions that go beyond the data (extrapolations and interpolations)

Lesson 2 Objectives:

a. To focus students’ attention on patterns of change in variables

b. To explore iterative and recursive change

c. To examine equations relating NOW and NEXT

d. To introduce and utilize the iteration capabilities of a graphing calculator or computer
software

Lesson 3 Objectives:

a To develop students’ ability to summarize patterns relating variables using rules

b. To utilize rules along with a graphing calculator or computer software

c. To illustrate the power of a rule and its capabilities when used with a graphing calculator
or computer software

d. To explore difficult questions relating variables by using rules and technology

Lesson 4 Objectives:
a. To begin to understand and develop an appreciation for the power of technology-

generated tables and graphs and the essential role of algebraic rules in providing
directions to a calculator or computer software
b. To recognize the connection between two-variable relationships and their graphs

211

List of Objectives (cont’d)

III. Unit 3 Objectives:

To recognize patterns in tables and graphs of data that are modeled well by linear equations
To write equations in the form y = a + bx to model linear patterns in graphs or numerical data
To use table, graph, or symbolic representations of linear models to answer questions about the
modeled situation: (1) Find y for a given x; (2) Find x for a given y (i.e., solve equations and
inequalities); and (3) Describe the rate at which y changes as x changes.

A.
B.
C.

l.

2.

3.

Lesson 1 Objectives:

a. To organize and interpret sets of data from real-world situations using a table, graph, and
equafion

b. To use a variety of methods to estimate the graph and equation of a line that fits a given
set of data

c. To draw and use a modeling line to predict the value of one variable given the value of
the other and to describe the rate at which one variable changes as the other changes

Lesson 2 Objectives:

:1. To construct a table of values from a given graph and examine these two representations
for common patterns

b. To write an equation of a line given its slope and y-intercept and interpret the meaning of
the equation, slope, and y-intercept in real-world problem situations

c. To write an equation of a line given two points on the line, explain the process for doing
so, and interpret the meaning of the equation in a real-world problem situation

(I. To produce a linear regression model using technology, and interpret the meaning

e. To describe how to use the numbers a and b to sketch the graphs of equations in the form
y = a + bx and write equations in this form given a graph or appropriate points of a graph

f. To explain the relationships among the graph, equation (y = a + bx), and table of values
for a linear model

g. To use and interpret the meanings of linear graphs, tables, rules, and related procedures in
real-world contexts to which they apply

Lesson 3 Objectives:

a. To write linear equations and inequalities that match important questions about linear
models or real-world situations

b. To solve linear equations and inequalities by inspection of appropriate graphs and tables
of (x,y) values, explain the methods of solution, and interpret the meaning of the solution
in the real-world context

c. To use the “undoing” and “balancing” methods to solve simple linear equations, explain
how to do so, and interpret the meaning of the solution in real-world contexts

d. To use the commutative property of addition and the distributive property of
multiplication over addition to rewrite linear models in equivalent forms

e. To test the equivalence of equations by comparing their tables and graphs

212

Table 8.7

Evaluation Inventory for Various Representations

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

VARIOUS REPRESENTATIONS -- _Graph_ 7 g 77 __ - 7 L AdLn Sara _Amyi Joy“ L[Terri Mon
5 ilntl - Tl LGraphing Time versus Temperature 0 ”1.4,. 3 7 4 ' 4 __ i 0 7 __4n _
6 In” - T1 iChoosing appropriate enseigssxssfi- _ r L4 it] 3-“, - .2 i _9 w“,
7 Lipil - Tl Choosing appropriate scales for axes __ _ 4 _ , .1 7 _ g g 7 0_ 7 i 77 77
.8 a int! -n jmng BEE???“ that do nos ran on the gricuinss __ _‘T _" _ 2.". _r 4. (LLB
Understanding where the origin is on the coordinate 1 4 2 2 0 2
9 Intl - T1 plane
_ , , , 4 4 4 4 4 4
I M — Tla Interpdaflnsferremts that deacttwgn theses: 99%.. _ ___ _ _.3
Using appropriate axes (x-axis for independent 4 3 4 4 4 . 4
2 Int2 - T3b variable) + L
+- —- -- --v _, — ———_L—— 7— 7 7 ' —4—77 4» — -» 7— ,_,
-, LM_ .,Cb_°°si118_ _ngrgpriaigenscafgrgzse s- _ I _L L _4__ _4__ __4_ r __4_ _ _ _4..
4 ma - T3b_LLChoosing7apprpp7riate scajes for axes -1 7 _ i 7‘: ‘1 Ligyi .
_ _ , , 4 4 4 4 I 4 4
LIE-W31?“ ADEMWIE that demtfejlm—mcgfllmesi __ _ _ , w _ ,_ _ r . _ ___L ._
1nt2 - T3e Graphed two functions on paper (using same set of E 3 4 1 4 l 1 2
#2 £55") 7% Laxe_s7 appropriately) k _i _ i __ _
Int2 - T3e ‘Graphed two functions on calculator by entering rules 0 0 O 0 0 0
15 (fxn) as Y=
Int3 - T2e2 Graphed two functions on calculator by entering rules 4 3 4 0 l 0 1
9 (fxn) asY= _ 4A—fi— __ L
Knowing that "y-intercept" describes the place where a 4 4 4 4 I 2 j
7 7 7l_ Llnt3 - "1:3Laifili7ne_crosses the y-axis _7 7 _ _7 _ ; _ _ ' 1
Identifying positive and negative y-intercepts, given 3 4 4 L 4 4 l ; l
H _ __lnt37-_T3a lines drawn on the coord plane __ ____L __ __ _ L __ 7
L_5__,_1m3_-I3LL ,Eorrectmiwtlaline with same y-imercept 4 s_4 i _i_ .. .‘i _ ' . _ '
Knowing that "slope" describes the uphill or downhill 4 4 4 3 3 1
3 lnt3 - T3a characteristic of a line 77 __ 7 __ __ __ i _ __ q
Identifying positive, negative, and zero slopes, given 3 3 4 I 4 3 2 l
4 lnt3 - T3a lines drawn on the coord plane __ ___ _7 7L ___ 7 ____ _JLA 7 7 77 i
6 mo - T3b Correctl 'd ' ' ' ' 4 ' 4 i 4 4 4 i 2
7_ 7 __g _ 7 7 7_ . 77 77% y Lentifying lines With same slope __L7___ _7_ i~ ;. 7__ L7 _w 7 7L _ _ 7,
. l i i l i i
' ! a r .
! Understanding the connection between the scatterplot 4 i 4 L 4 4 : 4 L l
g4 _Lhig-j‘la _ represepgamgdthc context situation _ ‘ ___ __ _ __ fl _ '_~ _7 _L _ 7 L
1 _- _ __ I- ____. _ i 1% __ __J___ -
_VARIOUS REPRESENTATIONS -— Table Adrn i Sara L Am L Joy L Terri ! Mon
W” ” 1 M ”’ ‘ ’ m“? T" ” ' "‘I ‘ " . I
,_|__1n}_1-_T_2___‘,C9£rsc_flmglsting 8 493131216 of data - r 4 «1‘5 -z_ “___ 1 , .4_ 1 ,f‘. __i__3z z
LUsing paired data to find a point of intersection of two 4 4 3 0 ' i j 2
3 lntl - Tl functions
3 lntl - T3 Using paired data to compare two functions 3 3 0 0 0 0
Identifying points on the TABLE where y>x (or y<x or 4 4 4 0 4 4
6 Int2 - le y=x)

 

 

 

 

213

Table B.7 (con’td)

 

Using a rule to find the y-values for a table of data

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

_ , 4 4 1 4 L 4 1 4 4
l Int2 -_ 7T3a [Washington DC tnp] _ - A 1 _ i 77 1
1 1 1
Int2 - T3e Entered two mics as Y= and examined table values for 4 2 0 L 4 1 NA 0
-1 1,3 A!!!) cemmrison -1- 1, 1 _ 1 1— __
Int2 - T3e Entered tt_vo sets of data into lists to make dual 0 4 . 0 1 0 ‘ 2 r 0
l4 (fxn) function representation ' _ 1 H'fl
16 Int2 - T3e Made her own written table 4 4 4 4 0 0
Int3-T2e1 1Entered tw__o rules as Y: and examined table values for 3 4 4 4 4 3
6 (fxn)_1comparison ______ __+_ _ _ _
lnt3 T2el 1Finding a point of intersection of two functions (Tl-83 4 4 1 4 3 4 3
7 (fxn) 1table) 1- 1 L L 1
1—"'“" *ws 7‘ “- -— " We ,. ' " “___" — W" L r ——* —r * ___—1*
1 _ ' __ _n_-1_ ILLC'”
VARIOUS REPRESENTATIONS - Rule (equation), explicit Adrn Sara A y 1 Joy 1 Terri i Mon
_ ——1 __ , _ . ,__+___ 1. Law,
1 1 1
0 0 4 4 4 4
4 11ml - T3 1USing an explicit formula to compare two functions 1
1Using a rule to find the y-values for a table of data 4 4 4 4 I 4 1 4
F_ ”71““an -_'1_'_3a__1_____[Washington DC trip] L _ ___L _ _ "r1 _ _
Writing a rule (formula) for a given situation [Hawaii 4 4 4 4 3 O
1.1 lest-id tan] 1 _ __;e_ _
Int2 - T3e Graphed one function on calculator by entering rule as 4 0 l O 0 0
W11 (fxn) Y= 1 _
lnt2 - T3e Entered two rules as Y= and examined table values for 4 2 0 4 NA 0
13 L(fxn) comparison L
1
11nt3 - T2c1 Entered two rules as Y= and examined table values for 3 4 4 4 4 1 3
6 1(fxn) comparison 1
, "T.” '7' _
lnt3 - T2e2 Graphed two functions on calculator by entering rules 4 3 4 0 L 0 1 l
9 (fxn) as Y= . 1 L f 1
- ———~» -- ~-— ‘1 L + ———— -t-———-. --—— ~
__1 1 1 1
--._-.____. _ _ -. _ _ __ . T‘_' 1‘ __ L_,
VARIOUS REIRESENTATIONS- Rule (equation), iterati_v__e _ *_ Adrn1 L7 Sara Amy .4 Joy ' Terri] Mon
.. . . 014.0'01010
L _ 5‘ lntl: T3 Denvmg NOW-NEXT equations for calculatlngvzfluesL _ A L __ _ 41—. 7 L1 7_ 7 7
Using the Answer Key on the T1-83 to quickly 4 0 1 4 0 V 4 1 0
6 Llntl - T“ calculate values iteratively (AnyTask) 1 1

 

 

 

 

 

 

214

 

 

Table B.8

Evaluation Inventory for Means of Solution

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

MEANS OF SOLUTION - Algorithms, Solving single equations Adrn Sara Amy Joy Terri Mon
int2 -
4 T2c/d Using the equations (rules) to find total earnings“ __‘4___ _H__4_ 4 4 _ 4 . _2 _
Using a given x-value to solve for Y --> y =
5 Int3 - T2d l(2x) + 4(x+4) + 4x 4 4 4 4 2 4
Using a given y—value to solve for X --> y = 1 ix .
3 lnt3 -'I“2b +12 4 _ L__4__ _ NA__L__4A‘L_4__L__4__
_i_2 Int3 - T2e3 Solving for X --> 8 + 10§= 12 + 9x 0 _4___ O 0 79 1 Q
l i 1
-7 _ ,__fi_ _ ..., _WL‘ ._ _
MEANS OF SOLUTION - Graphical, Points & Lines (sketched) Adi-ii Sara Amy _Lloy ’_ TEQLMon
5 Intl - Ti Graphing Time versus Temperature 0 3 4 4 0 4
Usmg graph to find a pornt of lntefSécTioioTtwo ' *fih“ '“3 _
4 Intl - Tl functions 0 3 2 4 l i
2 lnt2 - Tl a Drawing the y=x line accurately on the graph 4 4 3 3 4 3
Graphing Number versus Cost, using appropriate .
2 Int2 - T3b_Laxes 4 _ 3 4 4 1.1-43-1
lnt2 - T3e Graphed two functions on paper (using same set I 1
12 (gn)__ ___ of axes appropriately) 3 4 ___l__ _4 4_ __L __1*41 _ _2___
MEANS OF SOLUTION - Technology-assisted, graphlgg Adrn Sara Amy Joy Terri L __NI_0|_L
Entered one set of data into lists to make
6 Int2 - T3b _ _Scatterplot representation 4 4 _4 4 _ 4_ 3
Int2 - T3e Graphed one function on calculator by entering L
11 (fxn) gleasv= _4 0 Le 1.0 _1 . QM _ _Ao» L
Int2 - T3e Entered two sets of data into lists to make dual 1
i4 (fxn) function representation 0 4 0 0 2 O
Int2 31‘3e Graphed two functions on calculator by entering ‘ 1
15 (fxn) rules as Y= 0 0 0 0 0 0
Int3 - Using themegression line
6 le/c (y=a+bx) 4 4 3 3 3 3
[M3 - Using the linear regression model to predict a y- ,
7 Tib/c value for an x-value not in the data set 4 __33 _4_ ~_3_1L _ L _4_ -,_3___ .4
lnt3 - T2e2 Graphed two functions on calculator by entering 1
9 (fxn) rulesasv= 4__ 43114303141 1_1_
Finding a point of intersection of two fimctions
ELM-£2132 01-83 sragh) 3 L 3 3:33. (L- 2.13? L
_MEAN S OF SOLUTION - Technology-assisted, rule-generated t Adrn Sara Amy ng Terrl4 M_on
lnt2 - 'I‘3e Entered two rules as Y= and examined table
; (fxn) laiues for comparison 4 2 4 NA 0
nt - e Entered two ru g as Y= and examined table
6 (fxn) values figflcomparison 3 4 4 4 4 3
Int3 - T2e] Finding a point of intersection of two functions
7 (fxn) (Tl-83 table) 4 4 _ 4 __ 4-.} 4 _3_-3

 

 

215

Table B.8 (cont’d)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

M—EANS of soufi‘lon - Technology-assisted, conjecture & 1 1
i_nv_egfitrlgation ,, __ _ __ __ Adrn Sara Amy Joy 1 Terri 1 M091
1pattems as the functions extend twd x-approach-+- 1 1
19 Int2 - T3e Linnnity __ ____,1__3_,__ __ _4____ _NA 4 7 1JL 1311 1L
1191132; 136 ,1U~°>i;ig_th_6D-A83 to cheek 00199111993133: . .4 __ _ .2. _ 1.1% 1 4 1 NA L NA
. L -
T r l
MEANS 0i” SOLUTION :Tech—nongy—isslsted, Student: w ___ — 1— _—' 1 1 ‘ 1 #1 71* if 1‘
lave-"ed means i Adm ..-...Sm Amy ..M 1 Terri 1 M991
Using paired data to find a point of intersection of 1 1 ‘ 1 1
3 lntl - Tl two functions 1 4 4 1 3 o _ f l 1 __ __2_ _
3 Intl - T3 1Using paired data to compare two functions_____ 1 ._ _3__ "-1“ 3_ ___Q_ _ 0 __ __LQ'LL 0__ L
. jUsing an explicit formula to compare two L i
4 1lntl - T3 1functions ______ Li. L__ o__ 1-. ‘1____4__._t_ 4 ,1 4
1 1Deriving NOW-NEXT equations for calculating f L ' 1
5 'Intl -T3 values ' 0 4 1 0 I 0 O " O
' ‘ “iUsfiigEthe AnswerKéy on the 'I‘I-83 to qulfltly ’ "A
6 lntl - T" calculate values iteratively (Any Task) 4 0 4 0 4 0
l6 Int2 - T3e Made her own written table 4 4 1 4 4 1 O L 9 A
l 1
MEANS OF SOLUTION - Alterngte mus offsoiltlgion ___L _ . , Adrn Sara 1 Amy '1 . Joy 1 Terri 1 Mon
Using a rule to find the y-values for a table of data " ' 1
l lnt2 - T3a_L [Washington DC trip] _L ______________ _ y 4
lnt2 - T3e Entered two rules as Y= and examined table
L13_L_(fxp) ______ galues for comparison Complementary _‘ _ '2
16 Int2 - T3e Made her own written table 4_ L
Using the estfiated linear mo<fiai to precfict a y- w T
3 Int3 - Tla value for an x-value not in the data set 2
Int3 - Using the linear regression model—to. predict ay-‘f-
______7 _ £11319-“ _ value for an x-value not 39.1.11? dgiej—_ _ _ 4
Using a given y-value to solve for X -> y = 1 ix
3 lnt3 - T2b + 12 _4
Finding a point of intersection of two functions
10 lnt3 - T2e2 (Tl-83 gpaph) 44 _, 2

 

 

 

 

216

Table B.9 Evaluation Inventory for Levels of Abstraction

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1
LEVELS OF ABSTRACF‘ON - Using math to make sense or C Adr Sara Amy Joy Terri Mon
situation
_, _1____# ..1____ _ _ _# ___ n
2 lntl - T2 Recognizing that the answer has more than one case _ -~4 __p_ L *4 «p_ 4%” 3v“! _3
Recognizing that the answer to the question has more
2 lntl - T3 than one case 3 3 3 3 4 1
Using points where y>x to answer something about the
_ 5 lnt2- le overall data 4 __ _4_ __ 14__ _ ___4____ 1 ;_ __ ___l__
Using points where y<x to answer something about the
7 lnt2 - Tlc overall data 4 4 _4 p; _4_+ __9__11_ _ __
Using points where y=x to answer something about the 1
8 [rig—Tlc overalldata 4 o 41 __Q_ 1____3_‘ 1101
Being able to explain why two equation-values are the
__ 5 lnt2 - T2e same if their "constants" differ _i 7L 7 p4_ 4_ p 4 74+ 7 4
Being able to describe the relationship between the 3
7 lnt2 - T3c function variables 1» 4 4* _- 4g~ _4 $.33 __ ‘4 _
Predicting the shape of the graph for a function similar
10 lnt2 - T3d to one already graphed 4 4w 4 _ 4 , 4 1 ___NA _1
16 Int2 - T3e Made her own written ta_bl_e _ _ F 4 _ 4__ 4 __ p _4fl___ _9 __O__fi
l7 lnt2 - T3e Predicting a y-valuefpr an x-value not in the data set 4 4 _ _2__ ”Mi ‘4» NA +4-. 4 1
2| lnt2 - T3f Being able to describe how the two functions compare 4 4 4 4 4 3
Finding a linear model that is a good fit for the trend in
l lnt3 - Tla the data by eyeing with a clear niler 4 4 __3 _ 4 4 3
Using the estimated linear model to predict a y-value
3 Int3 - Tla for an x-value not in the data set 4 __4_ 4 2 _ 4 __l _
Understanding what the x- and y-values represent from
8 lnt3 - T2e] the problem context 4 4 .17 p 4_ A 2_ __4_ 4
Understanding what the x- and y-values represent from
11 lnt3 «- T2e2 the problem context 4 4 4 0 3 _ _1_Ofiw
LEVELS or ABSTRACTION - Mathematizing a situation 1“" S_'_':f1_ Amy J” 7:" M”
l lntl - Tl Vocabulary -- "At a rate of per ..." 4 4 4__ W «@1411 1 “34.3% _3 1 _ 1
Understanding that the question implies At what time I 1 1 1
2 lntl - Tl are the temps the same? 2 _ 4 4 _ 1 4“ 1‘1” 1 l _
Using graph to find a point of intersection of two 1
4 lntl - Tl functions 0 _ __3__ 2 4 _l_ _ I
5 lntl - Tl Graphing Time versus Temperature 0 3 4 4 __0 4
Understanding that the question implies Which option
1 Intl - T3 is more economical? 4 4 4 4 4 3
3 lnt2 - Tl a Understanding what the y=x line represents 3 2 3 l 3 l
Translating words into an algebraic
l Int2 - T2a/b representation/expression 4 4 4 1 4 2 ;2

 

217

 

Table B.9 (cont’d)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Y 1 1
2 Int2 - T23 Assigning letters to variables 4 4 4 1 4 1 4 1_4
._ T _M w - _ _L
9 lnt2 - T3d Used Washington rule as a model for Hawaii rule ? 4 A _ 4__p_ _ _"_ 31m 0___
lnt2 - T3e Graphed two functions on paper (using same set of 1
12 (fxn) am appropriately) 3 4 l 4 l 1 2
Understanding which line is the best fit and knowing
2 lnt3 -Tla why (when eyeing withaclear ruler) __2_ J ___4_ _ _ 2 _4‘1 _22
Using the TI-83 to find the Linear Regression line (y = 1 1
6 lnt3-lelc a+bx) _4__ _4_ __3 _ ; _3______31__§
1
Using the linear regression model to predict a y-value 1 1
_ _7__;ln_t3_ 5_le/c for an x-value not in the data set _4_1- 4__ _1 L. 1 3 . __4 _1 1 __1 3__fl
Translating words into an algebraic .
representation/expression (using 3 similar, worked-out
4 Int3 - T2e example) __ _ _4 __ m4 __ 1___4____ 1* _ _l __p _l _ L _ 3--
gyms or ABSTRACTION - Symbolic Abstraction ____ Mr 51ft} 143131“ ’9 1199150"
Recognizing that there are many places on graph (other
4 Int2 - T1 3 than data points) where y=x 0 0 0 l 0 0
3 lnt2 - T23 Writi_ng equivalent expressions/equations 4 3 4 2 0 0
Manipulating expressions/equations - applying the
l lnt3 - T23 distributive property 4 _4_ l l 1. _ L1 l _ a
Manipulating expressions/equations -- collecting &
2 lnt3 - T23 combining like terms __4 4 l 1_ fl“ _L
_2 “3%me - T2b Using a given y-value to solve for X -:> y = llx__-_I-_l_2_ 4 4 NA 4 _ _4 2 1 _4__
Using a given x-value to solve for Y --> y = l(2x) +
_ 5 lnt3 - T2d 4(x+4) + 4x 4 4 4 4 2 4
12 lnt3-T2e3 Solving forX--> 8+10x=12+9x 0 4 0 0 0
Using knowledge about the y-intercept value appearing
___Aplnt3 - T3e in y= -2 + 3x to plot a point at (0,-2) 4 l 1 ___4 .. _ 3 __ _(_)1_ ‘17 2
Using knowledge about the slope value appearing in y
8 Int3 - T3e = -2 + 3x to sketch an ”uphill” line thro y-int 3 2 _ _ __4 _ __ 3 *1 _ _Q __q _ l
LEVELS or ABSTRACITON - Symbolic Generalization Ad? 5"" Amy J0! Tm‘ M0"
9 lnt2 - Tl d Knowing what the data do/not suggest 4 2 l l 2 l
18 Int2 - T3e Understanding the infinite extent of a function 4 4 NA 4 NA 2
Making reasonable conjecture abt numerical patterns
19 lnt2 - T3e as fxns extend twd x-approach-+-infinity 3 4 N A 4 NA 1
Understanding what makes a good estimate and why .
__ 5 Int3 - Tla (cf. interpolating betw 30- and llO-story bldgs) l 1___ l _1 _ ‘___l ___1_ 1 _ 1
Understanding of what makes a good estimator and
8 Int3 - Tld why (lines where F.87 and r=.999998) l _ _l 4 _4__ F _L, 1 __l___
Understanding what C and D represent in an abstracted 1
9 lnt3-T33 linearequation --> y=C+Dx l _l___fi__.l_ _pkl 1__l 11_ l__
Knowing which of C or D is slope and which is y-
10 lnt3 - T33 intercept 4 l 1 l 2 l 1 l

 

 

 

 

 

218

Table B.10 Evaluation Inventory for Functions

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

EUNCTIONS- Linear Mr 13ra1An-y 1 11°11, LTorrl 1 Mon
11 111111 -T1 IUslnggraph to findapointotlnterseotion oftwo1 0 3 2 1 11 1 1
" 2'*~--r' a ‘ r” ,, in
1 Using paired data to find a point of intersection 1 1 1 1
31lnl1-T1 011111101“ 11 s 4 1 4 1 3 0 1 1 2
i 1 ‘
Undersnnding which line is the best fit and 1 ‘
2 lnt3-Tla knowingwhy(| ayelngwithal ruler) 1 2 1 4 4 2 1 4 2
, , _ 7 , ___, 1 ‘ _ 1 l, i
1 Using the estimated linear model to predict a y- j ‘
3 1lnt3-Tta valueforanx-valuenotinthedataset 4 1 4 1 4 2 I 4 1
., a _,_ m ' ,4, in; , , ,i 2
Understanding what makes a good estimate 1 1 1
5 1int3-Tta andwhy(cf.lnterpoiatlngbeM80-and110— 1 1 1 I 1 1 1 1
_,,‘ storybldas) 1 7 7 ,2, 1
IUnderstanding of what makes a good estimator i
8 "“3 ‘ T” and why (lines where r=.87 and amass) ‘ 1 1 ‘ ‘ ‘ 1 1
2, a, 1 .<)__..
e n - EnteredMasW and examined table 3 4 4 4 4 3
T2e1 (fxn) 1 values for comparison
‘ At
‘lnt3 - Finding a point of intersection of two tunctions
7 1T2e1 (fxn) (Tl—83 table) ‘ ‘ ‘ 3 4 3
Int3 -hed1mm on calculator by entering 4 3 4 0 O 1
T262 (fxn) rulies as Y:
7?, ,2, , , , a.__,_1, 22,242,,“
FUN1CTIONS— Non-Linear (inverse variation) Adr1 Sara 1 Amy Joy Terri Mon
11 11n12- T3e Graphed one function on calculator by entering I 4 1 0 1 1 0 0 0
(fxn 1 mice sY= 1 1 7 1 41
11112 .136 Graphed two functions on paper (using same 1
12 (fxn) aetofaxeeappropn ')..ateiy .ioComp mantary ‘ 3 1 4 1 4 1 2
_ 7 Strategy __
111121.“, Entered two rules as Y= and examined table 1 1
13 (txn valuestoroompenson...Complementary 4 2 o 4 NA 0
1) Strategy 1 __1
111 int2 Tee Entered two sets of data Into lists to make dual 0 1 ’ 0 o 2 o
1(fxn) functionlepresentation 7 111711 ___ 1_11 1
151int2- T36 Graphed two functions on calculator by entering 0 1 0 o 0 0 0
5(thin) miss as Y= 1 1
7 ,2 1 2.1,, 7,, 7 new
18 int2 - T30 Understanding the infinite extent of a function , 4 1 4 1 NA 4 NA 2
4 — W ., 74+ —
191nt2-Tae, ' *‘~‘ ‘ “_,' 1314 NA 4 NA 1
infinity 1 1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table B. 1] Evaluation Inventory for Number Re1atio1nships

 

NUMBER RELATIONSHIPS- Constant variation

 

3 [NZ - Tia Understanding what the y=x line represents

NUMBER RELATIONSHIPS - Inverse variation

 

 

1,,

,
Being able to dcscnbe the relationship between the 1

Adrn Sara 1 1‘Amy

3

 

7 ma T3e 1 ftmction variab1es 4 4 4 4 1 4 1 4

10 11111% T3d Predicting the shape ofthe graph for a function 1 4 11 11 1 4 4 NA
7 11similar to one already graphed 1 1 1

21 111112 T3 1. 1113:1113“ able to describe how the two functions 11 11 1 11 1 4 4 1 11

Adrn 1 Sara 1 Amy

Joy iTen-t Mon

,3 L;

Joy Terri 1 Mon

 

 

219

Interview 2 with Adrienne—Interview Memo, 1/22/02

Questions about the Core-Plus Course 1 class

Adrienne has been the most pleasantly-surprising interviewee this round with her
positive attitude and her successful performance on the math tasks; she truly has had the
best conceptual understanding of [the five] interviewees to date. (It was difficult to find
the edge of her understanding!) Her attitude survey shows all Agree or Strongly Agree
responses.

Adrienne reported that, since the last interview, the Core-Plus Course 1 class had
gotten “a lot easier” because she understands it so much better. She said it just made a lot
of sense, like it had not before. This definitely was borne out in the interview/tasks. She
didn’t really have an explanation for why things have come so easily to her this year,
especially in light of her having failed last year. Even her familiarity survey shows that
she didn’t understand most of the Unit 2 topics prior to studying them this semester in
Mr. Larson’s class.

Problem Solving Tasks

Task 1: Grades in English vs. grades in Math.

Adrienne showed excellent plotting and interpolating skills. She worked very
carefully, neatly, precisely -- definite math-community-membership on those counts. She
had an answer of “6” down originally for Part C because her (74, 74) point looks like it
falls below her y=x line; but, in discussing an alternate way of finding points for higher
grades in one subject than the other (using the table), Adrienne changed her answer to “5”
because she found the two points for which the math & English grades were the same.

Task 2: Sara the waitress.

Adrienne is the first one who did not use I and T or D (alliterative variables) for
their rules; instead, she chose y and x. This seemed to indicate loudly & clearly to me
Adrienne’s math-community-membership on this count as well!

The same thing about the hours seeming like they should be a variable but being a
given number in the Rule was again puzzling to Adrienne. After I helped out a tiny bit,
she figured it right out.

Her first attempt at the Rule was Y = 2.50 X, but in explaining it to me, she
changed it to Y = 2.50 + X. But she realized that still didn’t fit exactly with her
explanation. So then - all without my prompting — she finally settled on Y = 2.50 (8) +
X. She was going to put in a multiplication sign at first between the 2.50 and the 8, but
she chose the parentheses saying “This is what I usually use.” Another indicator of math-
community-membership (MCM)! especially because it has not occurred to any other
interviewees to use parentheses to symbolize the multiplication operation.

One final indicator of math-community-membership was what Adrienne decided
to write down for Part E. I verbally asked the question stated there (“How does the
amount that Sara earned on Sunday compare to the amount she earned on Saturday?”),
and Adrienne (had actually already) said they were the same. Then I queried her about
that with my probe “Someone else said Saturday’s income should be less because the tips
were less” but Adrienne immediately explained it and then said, “Is that what you want

220

Interview 2 with Adrienne—Interview Memo, 1/22/02 (cont’d)

me to put down?” I said yes, so she wrote “She worked less hours on Sunday than
Saturday.” The reason I thought of this as a MCM indicator was because Adrienne
seemed conscientious about filling out all the answers on the sheet. No other
interviewees that I recall have shown this; many seem not to care about writing stuff
down, or they view it as an obligation “if they have to” (Joy, for instance); but Adrienne
seems to have the disposition that this is for the important recording of known
information, being sure that all the answers are kept in the record kind of like a good
science lab student who will follow the protocol of recording all observations carefully
and precisely because she wants to behave properly as a scientist and not just because
(/when) she will get points for her grade for doing so.

Task 3: Senior class trip.

There were many fascinating aspects to watching Adrienne work on this task!
Adrienne exhibited MCM, I thought, by drawing the separating line between her entry of
133.33 and 100 on her table for Part A. Just one other interviewee (Sara?), so far, has
done this.

Also, Adrienne found the entries for the table in Part A by punching in 8000/n on
the calculator. So, the fact that she wrote 133.33 for 60 people also indicates MCM, in
my opinion. I think she even said “133 dollars and 33 cents ” as she wrote it! (Check
transcript.)

Adrienne was very adept in her use of the graphing calculator for this task. The
only place she faltered was in remembering to set up the STATPLOT so that the points
on the table would show. I helped her just a little with this.

As Adrienne read the text for Part D, it seemed fairly clear that she also possessed
adequate MCM to understand the use of the “c” and the “n” in the last sentence. Other
interviewees, just in their reading-aloud, have stumbled on these. Adrienne’s inflection
was just how a MC member would read it!

On the graph for Part B, Adrienne labeled the X-axis “cost per person” at first.
Then she corrected herself and simply crossed that out and wrote “numbers” instead.
MCM? (I 'm not sure if this was why or not.)

Her numbering both axes from zero and LABELING the origin with a zero
showed MCM to me.

Finally, it was most interesting when it came time to draw the graph for Part E. I
was feeling certain that Adrienne would extend the numbering of her y-axis from Part B,
because there was plenty of room. However, she instead drew a completely new graph in
an unused area of the graph paper. This was perhaps the one place where Adrienne did
not exhibit MCM (I would expect a MC member to automatically assume that you would
use the same set of axes to graph the second function. ).

I waited for her to finish without interrupting, but then I probed her about
comparing the two graphs. Then it occurred to her that she might put them on the same
set of axes for comparison. So she redid the graph and we pursued the rest of the
questions I’ve been using for all interviewees (but which don’t appear on my interview
protocol). I remember that Adrienne did well on these, but I don’t recall exactly how she

22]

Interview 2 with Adrienne—Interview Memo, 1/22/02 (cont’d)

answered (about whether the functions would ever eventually cross or meet up). CHECK
THE TRANSCRIPT FOR THIS. ~

I do see that, on the graphing calculator, Adrienne went as far as

X = 11400 Y1 = 1.7544 Y2=.70175
and that she was skipping by 1008 for the X-values (more comfortable than Joy, for
instance, who thought she might miss it by skipping by more than ls -- ?).

Other settings Adrienne used on the calculator:

WINDOW STAT-Edit
Xmin = 0 L1 = 20, 40, 60, 80, 100
Xmax = 100 L2 = 400, 200, 133.33, 100, 80
Xscl = 10
Ymin = 0
Ymax = 400
Yscl = 50

She also had both functions entered into the Y = window. Y1 = 20000/X and Y2 =
8000/X. That reminds me of how Adrienne explained to me, when she entered Y1, that
she would use “an X instead of u” even though she had written c = 20,000/n for the Rule
for Part D. Also a sign of MCM — being conscious of (and reporting that she was) using
an X for some other variable-letter.

Also, the fact that she wrote her Rule for Part D as it would be entered into the
calculator (using the slash for a division sign and writing the rule left to right instead of
writing it like the text showed it for Part A with 8000 OVER n) seems to indicate
mathematical/conceptual thinking-ahead-to-solving-the-problem and some problem-
solving-independent-thinking, instead of just following (more blindly?) after the model of
how the text did it in a previous part of the task.

222

Interview Three with Monica—Excerpt from Transcript of Task 2, 3/20/02

Interviewer:

Interviewer:

Monica:

Interviewer:

Monica:

Interviewer:

Monica:

Interviewer:

Monica:

Interviewer:

Monica:

Interviewer:

Monica:

Interviewer:

Monica:

Interviewer:

Monica:

Interviewer:

Alrighty. Now, second time around, they each took one turn. Now they
take a second turn. On their second turn, the equation for Jerome’s points
was equivalent, or the same as

Jerome’s points was equivalent to Y equals eight, eight plus ten X.
Okay.

And the equation for Lin’s points was equivalent to Y equals twelve plus
nine X. So they’re saving us the trouble of doing all this. Okay? They’re
just gonna give us a short cut. Before they roll the dice they wondered
whether there was some number on the dice that would make them tie.
Okay? So you’re supposed to find out the answer to their question and
there’s three, at least three different ways you can useo this. Okay? To
solve this problem, and two of these I’m pretty sure you did in class. You
might have also done [the Equation Method] and so we’re gonna try that.
Okay?

Uh huh.

Do you remember using your calculator to compare the values of two
equations in Mr. Harper’s class?

Like Stat and Enter, is that ~ ?

Yes. (pause) Well, yeah, you could use the Stat and Enter or you could
also use the table over here. Do you remember using that table like that?

Whoa.

If Stat and Enter is more familiar to you, use that. Okay?
Uh, yeah.

Alright.

No numbers. [The student is referring to the use of a rule with a variable
to generate a table as opposed to entering numbers into a lists.]

Okay, yeah, [the List Method (using Stat & Enter keys) is] the one where
you have to punch in the numbers.

Oh. I don’t know how to use that.
Okay. Let me help. Um, I think the table’s actually a little easier.
Yeah.

And so those numbers right there, one, two, three, four, five, six. Since
they tell us in the problem that he has, his points equal Y, are the same as
Y equals eight plus ten X.

(Monica enters 8 + 10X into Y1 in the Y: window.)

223

Interview Three with Monica—Excerpt from Transcript of Task 2, 3/20/02 (cont’d)

Monica:

Interviewer:

Monica:

Interviewer:

Monica:

Interviewer:

Monica:

Interviewer:

Monica:

Yeah, you can punch that into there. And on the next line [in the Y:
window] is [Lin’s equation].

(Monica enters 12 + 9X for Y2.)
Oh sorry. Use [the 2nd button for] the table.
Oh.

And it’ll actually tell you the points. So, if they roll a six, how much does
she get for points?

Sixty-eight.
And what does he get?
Sixty-six.

Okay. The question was, is there a number on the dice that would make
them tie?

(pause) Yep.
Which?

Foun

224

Adrienne’s work on Task 3 of Interview Two——Comparing two functions

Interviewer:
Adrienne:

Interviewer:

Adrienne:

Interviewer:

Adrienne:

Interviewer:

Adrienne:

Interviewer:

Adrienne:
Interviewer:
Adrienne:
Interviewer:
Adrienne:

Interviewer:

Adrienne:

Interviewer:

Adrienne:

Interviewer:

Adrienne:
Interviewer:
Adrienne:

OK. Now. When you had the graph over here [on the calculator]
M Hm.

THAT one. (pause) Yeah. It’s got your dots which represented the
Washington, DC stuff, and then it put this line on here that represented the
Hawaii stuff. And it looks to me like, eh, eh, like you had said, y’know, it
would be bigger.

M Hm.

But HERE [on the graphs drawn on paper], the two of these look the very
same to me. (pause) Do you know what I mean?

Because of the amount skipped between the twenty and forty? Or is the

Yeah, ‘cause if I cut this out [the Hawaii graph drawn at the top] and laid
it on top, I’m guessing those would be almost identical.

M Hm.

Y’know, like held ‘em up to the light and I’d have [points on the Hawaii
graph] right over the [points on the Washington DC graph].

M Hm.

So what do you think about that?
(pause) Mm, pretty much the same
OK.

ifyoulook.

OK, so, can you think of why these [on paper] look like they would be
identical if you laid one on top of the other? But here [on the calculator],
when they’re on the very same graph, this one looks bigger?

(pause) I’m not sure.

OK. (pause) One of the things I wondered about was, I was surprised
when you started way up here, but then you numbered these from zero to a
thousand.

Oh you

And I was wondering about starting with the zero down here. Is that
possible to do it that way?

M Hm.
OK, is there any reason why you decided to do it that way?
(laugh) No.

225

Adrienne’s work on Task 3 of Interview Two—Comparing two functions (cont’d)

Interviewer: OK, that’s fine! (laugh) ‘Cause I’ve had people do it both ways, but
you’re saying you could’ve used the same zero, four hundred,

Adrienne: Yeah.

Interviewer: and gone up to a thousand? (pause) Would that make the two of those,
then, look like it does here [on the calculator]?

Adrienne: (pause) erh

Interviewer: OK.

Adrienne: probably.

Interviewer: OK.

Adrienne: It’d be closer together and easier to compare
Interviewer: OK, alright.

Adrienne: that way.

226

Sara’s work on Task 1 of Interview One—Comparing two functions

1. At 3:00 P.M., the temperature is 86° F in Atlanta and is
decreasing at a rate of 3 degrees per hour.

At the same time, the temperature is 56° F in Fort Wayne and Is
increas'ngata rateof2degreesperhour.

When will the temperatures be the same?

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227

Adrienne’s work on Task 1 of Interview One—Comparing two functions

I. At 3:00 P.M., the temperature is 86° F in Atlanta and is
decreasing at a rate of 3 degrees per hour.

At the same time, the temperature is 56° F in Fort Wayne and is
increasing at a rate of 2 degrees per hour.

When will the temperatures be the satire?

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228

Amy’s work on Task 1 of Interview One—Comparing two functions

1. At 3:00 P.M., the temperature is 86° F in Atlanta and is
decreasing at a rate of 3 degrees per hour.

At the same time, the temperature is 56° F in Fort Wayne and is
increasing at a rate of 2 degrees per hour.

Whenwiilthetemperaturesbethesame?

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229

Joy’s work on Task 1 of Interview One—Comparing two functions

1. At 3:00 P.M., the temperature is 86° F in Atlanta and is
decreasing at a rate of 3 degrees per hour.
At the same time, the temperature is 56° F in Fort Wayne and is
increasing at a rate of 2 degrees per hour.

When will the temperatures be the same?
. .2:

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230

Adrienne’s work on Task 3 of Interview Two

Conjecturing about the functions c = 20000/n and c = 8000/n

Interviewer:
Adrienne:
Interviewer:

Adrienne:
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Adrienne:
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Adrienne:

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Adrienne:
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OK. Here’s my next question. I drew this little sketch here.
M Hm.

This, sometimes it makes sense to connect dots for graphs and sometimes
it doesn’t; and here’s one where it doesn’t. Pretend this is a graph of
people selling Girl Scout cookies.

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M Hm.

And, so, here’s you, and you sold ten. Here’s me, I sold five. Somebody
else sold five and somebody else sold eight. It wouldn’t really make sense
for us to connect those dots because they’re comPLETEly different
people.

Yeah.

In the case of THESE [senior trips data], would it make sense to connect
these dots, do you think?

(pause) It doesn’t really matter, you could because it’s at a downward
slope

OK.
and not up and down, up and down.

OK, OK. Umm, I’m also wondering about, you had talked at one point,
y’know, whether we would use the same numbers: twenty, forty, sixty,
eighty, a hundred?

M Hm.

Umm, and I asked, “Well, would it make a difference if you went twenty,
thirty, fifty, seventy, ninety?” Umm, is there a way, using this graph, to
tell what those numbers would be? Like if I wanted to know, I’ve got fifiy
students who want to go on the class trip, and we want to know the price
for Washington compared to Hawaii. (pause) Can you tell us how much
that would be?

Not exactly, but, uh (pause) precisely [sic] four hundred?
OK, and how did you figure

About four hundred.

that out? You’re estimating?

The space in between the forty and the sixty

OK.

231

Adrienne’s work on Task 3 of Interview Two

Conjecturing about the functions c = 20000/n and c = 8000/n (cont’d)

Adrienne:

Interviewer:

Adrienne:

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Adrienne:

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Adrienne:

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Adrienne:

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Adrienne:

Interviewer:

Go up [i.e., interpolating] and it, may be in between the forty and the sixty
dots.

OK.

And right there should be about

OK, and what do you think it would be for the Washington trip?
For what?

The Washington trip

Five hundred?

with the pencil dots.

Oh.

For fifty students, yeah.

(pause) About one hundred.

OK. And is there a way to check that?

On the calculator?

OK. So go ahead and see how close you came with your estimates.
(pause) Go by (pause) tens.

OK.

Starting at

So you’re just gonna change the table? And have it come up with?
Uh huh.

Alright.

(pause) Four hundred.

OK.

For the

That’s exactly what you said, isn’t it?

M Hm.

OK.

And I’m not sure how to check the, unless I put this rule in.
OK.

Forget what that was.

It’s on the other side.

Oh. (pause) Mm. Eighty

Could you figure it the same way that you figured these? ‘Cause these you
came up with without putting that table in. (pause) Remember that?

Yeah. Prob, yeah.
OK, how would you do that then?

232

Adrienne’s work on Task 3 of Interview Two

Conjecturing about the functions c = 20000/n and c = 8000/n (cont’d)

Adrienne:
Interviewer:
Adrienne:
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The same way it said to do, I wrote my rule.
OK.

I’d just type it in

OK. Alright.

the number.

And you can do it either way you wish.

(pause) Eight thousand divided by X. (pause) For fifty. (pause) That’d
be four hundred. Oh! (pause) One sixty

OK. (pause) Alrighty, so four hundred for the Hawaii trip, a hundred and
sixty
M Hm.

for the Washington trip. (pause) Alright. OK, now my next question
brings those two together: If you were to draw the connecting lines for
these dots, would that help you with those values, like if I wanted to know
fifiy students? If I wanted to know ninety students? Something like that.

(pause) Not, kinda, but not really ‘cause you’re still gonna have to go up
and find where the line hits

OK.

that.

OK, alright.
But yeah.

Alright. And another question about connecting those. If you were to
connect those dots, would you connect, would it make sense to connect
them with straight lines, y’know, like to use a ruler like this, y’know
[demonstrates what she means with ruler, miming drawing the lines], and
then, y’know, connect ‘em like this [more demo], etc.?

No.

What would you do instead?

Just draw openly, for it, with your hand

OK, why don’t you

use a curve

OK, why don’t you pick one of those and see how well you can do
I’ll do ~ ~

OK. (laugh)

(pause while student sketches curve through points) Dot that’s wrong.
OK, alright, so you’re just trying to draw a smooth curve

M Hm.

that connects all of those. Alright.

233

Adrienne’s work on Task 3 of Interview Two

Conjecturing about the functions c = 20000/n and c = 8000/n (cont’d)

Adrienne:
Interviewer:
Adrienne:
Interviewer:

Adrienne:
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Adrienne:
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This one [the point (100, 80)] doesn’t really look
OK, it doesn’t follow that same
Nh nh.

curve, you’re thinking? And it may be because, y’know, there’s such a
tiny amount in there

Yeah.

Um, y’know, that this might not be just exactly right, y’know and, which
is always a problem (laugh), y’know when you’re trying to graph
something with a lot of numbers by hand.

OK! One more set of questions for you! Um, well, lemme have you go
ahead and -- I told you I was gonna have you only do one — but I’ll make
you do the other one

Both of them?

Yeah, ‘cause that’s what my next question relates to.

(pause while student sketches curve) I’m not that good at it.

(laugh) It looks you’re pretty good at it. (laugh) I think that looks good.

Alright. If we continued to go on, let’s say we went to two hundred
students, three hundred students, four hundred students, how would these
two curves continue on? Can you predict?

They’d probably both come down at the same, mm, m mm (pause) I’m
not sure. Probably about the same amount? Maybe, umrn, after a while
(pause) it’d be the, like (pause) (laugh) I’m not sure.

OK, by the same amount you mean, like, they’re about one finger width
apart

Yeah.

And you think they would continue to be about one finger width apart as
they go?

(pause) M (longer pause) Nnno. I’m not sure. (laugh)

OK. (pause) Is there a way that you could use the calculator to check it
out?

Probably.
OK.

For, if, see if they’ll put two [punches some buttons] Wait, lemme go
back. [more punching]

So you’ll go back to that table?

M Hm.

And have it

~ ~ for two hundred, and they’re a hundred apart.

234

Adrienne’s work on Task 3 of Interview Two

Conjecturing about the functions c = 20000/n and c = 8000/n (cont’d)

Interviewer:
Adrienne:
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Adrienne:

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OK.
(pause) For the Hawaii it’d be a hundred, and for Washington forty. And
OK.

About forty apart there, when they were sixty apart there. Then for four
hundred, there’s only thirty

OK.

Five hundred there’s twenty, twenty-four.

OK.

And, then there’s twenty. And (pause) seventeen,
OK.

just about, not counting that.

OK. Alright.

And then fifteen ~

OK, so how would those look on the graph then? Does that help you?
Coming closer. They come closer together.

OK, so as they, they go they get closer and closer.
M Hm.

Do you think there would be a point — y’know, if we had all kinds of
paper, just kept going — umm, where those two would ever meet up?

(pause) Yeah.

OK.

I don’t know where, but they probably would,

OK.

‘cause this one gets shorter, by more

OK.

The Hawaii ones get shorter by more than the Washington one.
OK. Now is there a way you could use the calculator to try and see where
they might meet up?

Yeah, you can just use the, urn (pause)

And then how are you gonna know when they meet up?

It’ll be the same number.

Where?

Going straight across [on the calculator’s table display].

In those two, the Y1 and the Y2 column?

M Hm.

OK.

235

Adrienne’s work on Task 3 of Interview Two

Conjecturing about the functions c = 20000/n and c = 8000/n (cont’d)

Adrienne:
Interviewer:

Adrienne:
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(pause while working on calculator)

I promise this is my last question. (laugh) But you were doing so well, I
thought I’d go further. (laugh) Plus we had some time. (pause) How
close are they getting? I can’t hardly ~

They’re like three apart

OK.

for each one.

(pause while student continues with calculator) That’s really cheap, isn’t
it? (laugh) I’m thinking like a DOLLAR or something to go on the trip,
of course you’re putting up with fifty-nine hundred other people! (both
laugh)

(pause) I don’t know!

Are they not coming together?

Nh nh.

How close are they getting?

There’s like two dollars

OK.

About one-something in between.

OK. (pause) Do you think it’s possible they’ll NEVER MEET?
Maybe! ‘Cause no matter what, that one’s still gonna keep going down.
OK.

I don’t know.

OK. (pause) Well you did very well. (pause) The answer is they aren’t
ever gonna meet.

They aren’t?
(laugh) No!
Yeah, they ~ ~

And you’re exactly right because this one keeps going down and THIS
one keeps going down. They may get CLOser,

Yeah.

buuut, y’know, you could take a microscope to that graph, y’know, and
there’d still be just a little tiny space between

Yeah.

And, the other way of kind of thinking or reasoning through that is that
this is now down to like seventy cents

M Hm.

per person to go, but, y’know, the question that one of my interviewees
and I were talking about was whether it would ever go below the zero.

236

Adrienne’s work on Task 3 of Interview Two

Conjecturing about the functions c = 20000/n and c = 8000/n (cont’d)

Adrienne:

Interviewer:

Adrienne:

Interviewer:

Adrienne:

Interviewer:

Adrienne:

Interviewer:

Adrienne:

Interviewer:

Adrienne:

Well, as long as you have a certain amount of cost for the trip, and a
certain number of people going, you can’t take the cost divided by the
people and end up with a cost, y’know, per person

(laugh) Where they ~ ~
is zero.
Yeah.

Yeah. And so that’s how we were thinking about that, too, that it could get
smaller and smaller and maybe get to pennies, or tenths of pennies, or
something like that

M Hm.

but it’s NEVer gonna get to below zero, and no matter how many, like
y’know, if you have a twenty-thousand-dollar trip and an eight-thousand-
dollar trip, there’s still gonna be a difference in price

Yeah.

Y’know, even if I said let’s take a MILLION people. (laugh) Y’know,
well, the total cost is a little more, so the cost per person is gonna be a
little more

OK.

Y’know, for even the Hawaii travelers. (pause)
You did VERY well on that! (laugh)

(laugh) Thank you.

237

Sara’s work on Task 3 of Interview Two

Conjecturing about the functions c = 20000/n and c = 8000/n

Interviewer:

Sara:
Interviewer:

Sara:
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8:

Interviewer:

Sara:

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Sara:

Okay. Is it correct, would you say, to connect the dots? I mean is that
valid? Because I asked somebody else this, there’s some graphs where it
just doesn’t make any sense at all. Let’s say we’re selling Girl Scout
cookies, and these are the number of boxes so five and ten and here’s me
and I’ve sold ten and here’s you over here and you sold five and a couple
of other students. Now it wouldn’t make any sense at all, you know what
I mean

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Yeah.

to connect those dots? Does it make sense [for the senior class trips] to
connect the dots or is it more like [the Girl Scout cookie graph]?
Actually it probably wouldn’t matter whether you connect the dots or not
Okay.

cause that just helps you see the dots better, I think.

Okay. Okay. Alright. Umm, what if, one of your first questions you
asked me was if we were going to use the same numbers here. Could you
still get your results and stuff and be able to tell me how the two trips
compared and stuff? Urn, or even do them on the calculator at the same
time if we had used different numbers like ten and thirty, fifty, you know
that kind of thing?

That would have worked

Okay.

too,

Okay.

also as well.

Okay. So I’m wondering if the line helps tell you anything about those
other points on the graph.

Yes it does. I didn’t realize that before, but yes it does.

Okay. So, go ahead.

It would go through all the numbers like say ~ ~ I’ll just put a fake one in.
Like it goes through that or something.

Okay.

It would go through the points of all the other numbers.

Okay. So based on that you, what would you predict, um, it would cost
for fifty students?

For fifiy students it’d cost about a hundred and sixty-five

Okay.

to a hundred and seventy.

238

Sara’s work on Task 3 of Interview Two

Conjecturing about the functions c = 20000/n and c = 8000/n (cont’d)

Interviewer:
Sara:
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Sara:

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Sara:
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Sara:
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And that’s, uh, for which trip? (pause) 'Cause

For the New York, or the Washington DC.

Okay. And how much would it cost them to go to Hawaii if there were
fifty kids going?

(pause, while Sara sketches a line on the graph connecting the points)
About four hundred and twenty.

Okay. Alright. Now when you started sketching the line you started all the
way up here [from the highest point on the graph] rather than just draw a
straight line between the two why did you do that?

I don’t know.

Okay. Now are those suppose to be straight lines between the two? or are
they curving? because one of the things you said

They’re kind of curving.

umm, you described this as non-linear decreasing. (laughs) Which was
correct. Umm, and so I am wondering about that word, non-linear. Do
you know what that refers to?

It means not a straight line.

Okay.

That should be kind of curved.

So that would be curvy?

Yeah,

Okay.

Just a little bit.

Okay.

So that would be about four hundred,

Okay. Alright.

a little over maybe

Alright. Well and I was thinking it made sense, you know, that I was, I do
the very same thing you know if I’m gonna try to connect these dots with
a curve I, you know, start at the beginning so I can kind of get a feel for,
y'know, how that thing is gonna curve. So I think that made sense. I was
just wondering if that was

Yeah.

in your thinking at all. (laugh) Alright. I think we have one more question
to go. Umm, explain how the cost for the Hawaii trip compares to the cost
of the trip to Washington.

It would be quite a bit more.

Okay. In all cases?

In all cases because there’s a difference in the overall price

Okay.

and the one to DC would be less because the overall all price is less
Okay.

than the Hawaii one.

239

Sara’s work on Task 3 of Interview Two

Conjecturing about the functions c = 20000/n and c = 8000/n (cont’d)

Interviewer:

Sara:
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Sara:
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Sara:
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Sara:
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Interviewer:
Sara:

Okay. But I noticed how you commented about there being a huge jump
between here and here and when you said that, I noticed how there also is
a big jump between the difference, like for these twenty people it’s
REALLY different, y'know, in cost, like a thousand compared to four
hundred but when you get down to here (laugh) the difference isn’t so
much.

Yeah.

Like I was thinking, y'know, I might not as hard a time talking the parents
into going to Hawaii instead of Washington, DC if I we had a hundred
kids going because the amounts aren’t as different but here there’s no way
I could make that case.

Yeah.

And why do you think that is that, uh, when you’ve got fewer students its
REALLY a big difference?

Because there’s a lot, there's a big difference in the cost, the overall cost
Okay. Okay.

there’s twelve thousand dollars’ difference.

Okay.

and it would be more for each student.

Okay. Alright. Do you think there is ever a point, like of we just
continued to go on and on and on, urn, would those cost ever meet up, do
you think?

I don’t think so.

Okay. Why not?

Because there would al (hesitation) Well actually I think there might,
because these ones, cause there's, they're bigger then smaller and smaller.
Okay.

and I'm sure they would meet somewhere way along the line.

Okay. Alright. Do you have just a ballpark guess how many people you
might have to take to get those to meet up?

(pause, writing)

Okay tell me what you did there and why you did that?

I seen how many over it was

Okay.

and how many down it was

Okay.

And I’m gonna go here

Okay.

and see it again [Sara is approximating the slope between different sets of
two consecutive points]

Okay.

Umm, (pause) over the same amount, then down there's one, two, three
and a half

240

Sara’s work on Task 3 of Interview Two

Conjecturing about the functions c = 20000/n and c = 8000/n (cont’d)

Interviewer:
Sara:
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Sara:
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Sara:
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Sara:

Okay.

and we go over and then down two

Okay.

and probably over and down so it would be point five

Okay.

so it would be point five down (pause) and that would be (whispering,
counting) one, two, three, four. (pause) It would be right here.
Okay.

(pause) Um, I don’t know. I don’t think they would

Okay.

because you can’t get any smaller than zero. (pause) Because the next one
would be right here.

Oh I see, the difference that you’re decreasing it by.

M Hm, because that one went down three point five

Okay.

and then take away three point five minus one point five was two
Okay.

and that one was exactly two

Okay.

Then you take another one point five away

Uh hu. I see.

 

 

(talking over each other) and so if you took the point five here minus
the one point five it's gonna, alright.

be a negative.

Okay. That’s excellent thinking. Alright, umm, is there, do you know of
a way for the calculator to show you those numbers or the graphed non,
non line (laugh) the curve, going way out without actually having to sit
there and punch in all the numbers on your list?

No.

Okay. I wanna show you. See if this sounds familiar. Oops, if you, let’s
turn the plots off. Oop, too far. (pause) There we go. If you go to this Y
equals window (pause) and you use your

Okay.

rule. Does that ring a bell?

(pause) You can’t put n in the calculator so you have to make it like Y.
Okay. Okay.

That’s a Y ~

And that makes sense doesn’t it because you’ve got an X here and you’ve
got a Y here?

M Hm.

Okay.

Okay, then you would go to your tables, no Table Set, and you start it at

241

Sara’s work on Task 3 of Interview Two

Conjecturing about the functions c = 20000/n and c = 8000/n (cont’d)

Interviewer:

Sara:

Interviewer:

Sara:

Interviewer:

Sara:

Interviewer:

Sara:

Interviewer:

Sara:

Interviewer:

Sara:

Interviewer:

Sara:

Interviewer:

Sara:

Interviewer:

Sara:

Interviewer:

Sara:

Interviewer:

Sara:

Interviewer:

Sara:

Interviewer:

zero and go up by one. (pause) I’ll go by five.

Okay.

And, and you would go to your table. (pause) And zero says error because
there is no cost for zero students.

(laugh) Okay, uh huh.

And then you would, (pause) now what am I looking for?

So okay, so the line that you are on what does that tell me there?

That for ninety students it costs two hundred and twenty-two dollars.
Okay. Now can you, what we were trying to figure out is if they’re ever
going to meet up.

Okay.

So we put in this one. Now we want that one. And is there a way to get
all of those showing so that we could, like, compare 'em?

That would be the closest one right there. I don’t think that they would
ever meet up because there’s, they would never meet up, because those
two right there are the closest.

Okay. Where you got, what is that point?

One forty and Y was one hundred and forty two point eight six

[Here, the student is erroneously comparing the X-value and the Yl-value
of the same function to determine whether the two functions, Y1 and Y2,
will ever intersect]

Okay. And what does that stand for? The one forty and the one forty two
point six?

The one forty is how many students would go, and the one forty two point
eighty-six is the cost, is how much it would be.

Okay. For what?

The Hawaii trip.

Okay. How much would it be for the Washington, DC trip? For one
hundred and forty kids?

(talking as she punches into the calculator) Y equals, no, clear this, Y
equals eight thousand divided by next, (pause) one go up by five

Okay.

and (pause), that’s the closest one, ninety one students at the cost of eighty
seven dollars and ninety hun,ninety-two cents.

[Again, the student is erroneously comparing the X-value and the Y2-
value of the same function to determine whether the two functions, Y1 and
Y2, will ever intersect]

Okay. So when you say the closest one where that number and that
number are pretty similar.

M Hm

Alrighty.

But they will never meet up.

Okay.

242

UJE'.‘

Joy’s work on Task 3 of Interview Two
Conj ecturing about the functions c = 20000/n and c = 8000/n

Okay. Alright, um, I have a question about connecting those dots.
Yeah.

Some graphs, it makes sense to connect the dots and others it doesn’t.
Yes.

Okay. For example, let’s say I have this graph and it's people selling Girl Scout
cookies.

3, be)“; ][ o

r O .

 

 

A A s M
Uhhuh.

And the first person sold ten boxes, the next one sold five. That one sold five, that
sold about eight. It wouldn’t really make sense to connect those dots.

Yeah, I know.
Would it make sense to connect the dots on those graphs?

Mmm. (pause) I guess, if you wanted it to. It can but (pause) mm, yeah. It would
make sense. '

Okay. Would it help for any reason?
(pause) Mmm, help for any reason?
Yeah.

(pause) I don’t think so.

Okay. What if you wanted to find out how much it would cost for thirty seniors to go
either to Hawaii or to Washington, DC?

Then you could see if you made a line through it . (laughs)

Okay. So, would the line help in that case?

Yeah.

Alright. And how would you use the line to determine what it would cost?
You'd look, you'd, like go where you, from where thirty would be

Okay.

then go up and see where it meets the line.

Okay. Alright. What would you estimate for the pencil ones?

I accidentally put that forty,

Okay.

oh well.

So what would you estimate the cost would be for thirty students to go to

243

.V.’

w—‘m—‘m-‘w—‘w

Joy’s work on Task 3 of Interview Two
Conjecturing about the functions c = 20000/n and c = 8000/n (cont’d)

Washington, DC?

Umm. (pause) I’m gonna make a line here.

Okay.

Does it have to follow the dots? Or does it

That’s up to you. What do you think?

I don’t know, I guess. How ‘bout yeah?

Okay.

Okay. Thirty it would cost about two hundred and sixty.
Okay. Is there a way to check that to see if that’s right?
Umm, yes there is.

Okay how?

You could put (pauses to pick up the calculator) eight thousand divided by thirty.
What’d you get?

Two hundred and sixty-six point six.

Ooh, pretty good. Okay. Umm, let’s see. Almost out of questions. Ibet you’re
happy.

Oh joy.

(laugh) Oh yeah, explain how the, it's right here on the thing, explain how the cost of
the Hawaii trip compares to the cost of the Washington, DC trip.

They follow the same pattern as far as, like, the numbers decreasing but the Hawaii
trip costs more.

Okay. Umm, you see how these lines are, like, decreasing as they go?
Yeah.

And if you connected both of those,

Yeah.

Do you think there would ever be a point at which they came to, they crossed?
(pause) You know what I'm saying?

They might meet up at the same point.

Okay. You think they would, or they wouldn’t, or you don’t know?
Probably.

Alright. Where would you guess that to be?

Idunno. Ithink~~~

Like fairly near by or way out?

Mmm, (pause) Mmm, near by?

Okay. Go ahead and show me how you’d find out.

Can put it in Y equals, I think. (pause) Hmm, where is that? Then you go to second
graph, and they're both on here.

244

Vii-J

(Ill—J

Joy’s work on Task 3 of Interview Two
Conjecturing about the functions c = 20000/n and c = 8000/n (cont’d)

Mmmmm.

Then you see where they equal the same. I think.
Okay.

Maybe I did that wrong. But that’s okay.

Looks good to me. (pause) So you’re looking for which two to equal the same?
Yup.

The, these two? The Y1 and the Y2? Or these two?
Hold on. (laugh) I’m not sure.

Okay.

Oh, the Y1 and the Y2.

Okay.

(pause) Okay that’s the top of the other thing. Maybe I do not want to go by twenties
there.

Okay.
Start at two hundred and forty. Why on earth would I do that?
I think its just because you scrolled down the list a ways and that’s were it was.

Aye. (pause) What do I want to start at? How about, well we know it’s not there by
one hundred (pause) so

Okay.

How ‘bout ones? Why the heck not? (long pause as Joy scrolls down the table on the
calculator)

Are they getting closer?

Mmm, not really. They're getting close ~

Okay. (laugh)

(pause) Mmm, I’m not sure they will.

Okay. How far have you gone there? (pause) What if you jumped way way up?

Then I would get confused. [Student does not want to scroll through the table by X-
value increments greater than 1.]

Oh, okay. (laugh)

(pause) Because sometimes if you do it way up then you get, like, you see where it,
above where it matches.

Alter it’s crossed.

And miss it.

Okay. Okay.

(pause) like the same thing when you’re doing things that when they turn positive
Uh huh.

like when it hits bottom

245

S?

H
a.

Joy’s work on Task 3 of Interview Two
Conjecturing about the functions c = 20000/n and c = 8000/n (cont’d)

Uh huh.

you can't tell, but it’s already at a negative [describing to a critical value of a
function] .

Okay.

so you know whatever it did, you just don’t know when.
Okay.

I don’t think its going to happen.

No. Where are you at now?

Seventy.

Are you getting any closer than you have been?

Ahh, sixty-five and twenty-six, twenty-five and sixty-four, mmm, maybe. Perhaps.
(pause) Yeah, they’re getting closer.

Okay. (long pause) What’d you think?

I'm gonna change this real quick.

What'd you get to, four fourteen?

Yeah.

Okay.

So I'm gonna put, five, fives.

Okay.

Okay that would be ~ ~? I don’t know. (pause, while scrolling farther)
That’s getting’ cheap isn’t it? (laugh) Fifteen dollars to go then?
Yeah.

to Washington DC. It’d be a deal.

Yeah probably. (pause) Hm, I’m just thinking it’s not gonna cross. (pause) Although
it probably did somewhere.

You think this is ever gonna get to zero?
Yeah.

So it would be free to go?

Eventually (pause) it’ll get to zero.
Okay.

But will it get to zero before it crosses (pause) ~ trying to tell. (long pause) I don’t
think they’re gonna cross.

Okay. Where are you at now?
Eighteen and seven. [These are the Y-values for Y2 and Y1, respectively]

Okay. (laugh) That’s a lot closer than it was. That’s with [X-values of] what, eleven
hundred fifteen, sixteen?

They might cross at, like, negative something.

246

H
on

$4

Joy’s work on Task 3 of Interview Two
Conjecturing about the functions c = 20000/n and c = 8000/n (cont’d)

Okay. Alright, so is that the point at which you give up?
Yeah, perhaps.

(laugh) Alright. That was the most persistent of anybody I’ve asked. Very good.
(laugh)

Does it cross? [Joy is asking the interviewer for the answer]

No, it doesn’t. It actually never reaches zero, either. But it gets soooooo tiny you can
hardly tell ~ ~ ~.

It never reaches zero, it goes by ~ ?
It would never reach zero.
Never?

Nh nh. I mean it would just, it'd go to fractions of cents and stuff. Because if you
took

Point zero?

Well, yeah it would go to, m hm, zero point something, but it would never get to the
zero or below the zero because you would always have a certain amount of money

divided by a certain number of people, and you’re never gonna get zero. You know
what I’m saying?

Mhm.

Like if you had twenty thousand divided by twenty thousand it’d be down to a dollar
apiece; or if you had forty thousand people it’d be down to fifty cents a piece.
(pause) It’s a, it's a looong ways out there to find. (laughs) But it actually gets closer
and closer to zero and never reaches. Is that what you expected?

Mmm. I dunno.
(laugh) Well you did very well.

247

llx ti: =' $5

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y. u... ALA

Figure 3.1
Correct circle diagram

248

BIBLIOGRAPHY

249

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