The Effect of Technological Representations on Developmental Mathematics Students?
Understanding of Functions
by
Lauretta Elliott Garrett
A dissertation submitted to the Graduate Faculty of
Auburn University
in partial fulfillment of the
requirements for the Degree of
Doctor of Philosophy
Auburn, Alabama
August 9, 2010
Keywords: mathematics, education, technology, representation, adult learners
Approved by
W. Gary Martin, Chair, Professor of Mathematics Education
Marilyn E. Strutchens, Professor of Mathematics Education
Daniel J. Henry, Assistant Professor of Educational Foundations, Leadership and Technology
Stephen E. Stuckwisch, Assistant Professor of Mathematics
ii
Abstract
The use of technology in mathematics education has been strongly encouraged by
the National Council of Teachers of Mathematics (NCTM, 2000) and the American
Association of Two-year Colleges (AMATYC, 2006). Researchers have envisioned
technology?s potential in grand ways, including democratization of access to higher
mathematics (Kaput, 1994). There are challenges to the realization of that dream. For
example, innovation in technological advances often outpaces the evaluation of how
those advances can be best applied (Epper & Baker, 2009).The need for improved use of
technology in adult developmental mathematics education has been documented
(Caverly, Collins, DeMarais, Otte, & Thomas, 2000; Epper & Baker, 2009).
At the same time, adult developmental mathematics students? need for support
and help to realize their educational dreams is a vital current issue (Bryk & Treisman,
2010). This study seeks to provide insight into how the use of mathematics technology
affects the internal mathematical representations possessed by adult developmental
mathematics students. It is hoped that such insight may provide teachers of adult
developmental mathematics students with research based understanding which will aid
them in incorporating the use of technology.
Open recruitment was done on the campus of a mid-sized university in the
southern United States. One subject was interviewed 7 times and then a second subject
was interviewed 6 times. Each interview was video taped with three feeds to capture the
iii
subjects? interactions with both paper and technology and to record the subject?s
movement and facial expressions. Qualitative analysis was done with the aid of Atlas.ti
software during and after data collection. Each case was considered separately,
compared and contrasted and merged results were also considered. Results suggest ways
in which technology can impact student thinking.
iv
Acknowledgments
First and foremost I wish to thank my Heavenly Father for encouraging me,
sustaining me, specifically and consistently answering my prayers, and giving me so
many wonderful blessings which are too numerous to name here. Without him I am
nothing.
I would also have very little to show for myself without the support of my
immediate family. I thank my husband Charley for always encouraging me in whatever it
is I wish to pursue, and for all of his amazing love and support. None of this would be
possible without him. I thank my children for being the wonderful people they are, for
their wonderful listening ears and all of their encouragement: Dana McKeen, Jennifer
Garrett, Christopher Garrett, and Joseph Garrett. I also thank Benjamin Garrett for the
example his life was to me of doing your best with whatever you are given and for
providing his unique light to our family and all who knew him.
I have had the opportunity to work with what must be two of the most devoted
and caring professionals in mathematics education, Dr. W. Gary Martin, and Dr. Marilyn
E. Strutchens. I cannot thank them enough for their examples and continued support and
caring. Specific thanks to Dr. Martin for guiding me through the process of sorting my
thoughts and finding my way through my own weaknesses. His never failing ability to
cut through the non-essential and see a greater vision was a joy to experience. His
encouragement went beyond the realm of the academic as did Dr. Strutchens? support
v
during this time. Thanks to Dr. Daniel J. Henry for helping me to understand what
qualitative research is all about, encouraging my initial attempts, and listening patiently
to my concerns and questions. Thanks are also due to Dr. Stephen Stuckwisch for his
cheerful encouragement, particularly as I went forth to study a higher level of
mathematics.
There are others without whom this study would not have been completed.
Thanks are due to Pam Ketterlinus for helping me to get through this with her wonderful
support and special consideration. Also my thanks must be given Dr. Cindy Henning, Dr.
Tim Howard, Mr. Hassan Hassani, Dr. Terry Irvin, and Dr. Eugene Ionescu for their
special support. Particular thanks is also due to Susan Anderson who answered my
numerous questions with such great patience and encouragement.
I have had many wonderful teachers during my life and must mention them as
well. It was wonderful to sit in the mathematics classrooms of Dr. Phil Zenor, Dr. Dean
Hoffman, Dr. Pete Johnson, Dr. Randall Holmes, and Dr. John Hampson. Each of them
contributed something wonderful to my mathematics education. Special recognition
should be given to Ms. Beverly Davis and Dr. Mary Lindquist who were guides and
inspirations as I ventured into the field of mathematics education. I wish also to recognize
Mr. Ken McMeans, who embodied enthusiasm for his subject, and took his students
beyond the borders of his assigned curriculum. Ms. Doris English showed what a high
school classroom really should be ? to enter it was to desire to learn and contribute. I also
recognize Mr. Gerald Kievman for taking a special interest in me and showing that you
really can give individual attention to students in a meaningful way in our modern
academic system.
vi
I cannot fail to thank other wonderful people with whom I have interacted during
this time, and who provided their special kind of support. Thanks are due to Lisa Ross,
Mary Johnson, Gayle Herrington, Dr. April Parker, Dr. Lora Joseph, Dr. Mary Alice
Smeal, Carol Gudauskas, Clarisa Williams, Charmaine Cureton, Justin Yeager, Anna
Wan Brice, and Dr. Steven Brown. You have been examples, listening ears, and moral
supports. Thanks also to Elaine Prust for always being so cheerful and helpful. Bishop
Gordon Murphy and Bishop Kendall Ence provided their listening ears on many
occasions. Thanks also to Janice Grover, Carol Reid, Sue Funk, Linda Lenhard, Courtney
Pierce, Ginger Mendoza, Deirdre Davis, Jana Martin, Denise Kimbrell, and Elizabeth
Holloway.
Finally I want to thank my parents, Norman Paul Elliott (1926-2002) and
Elizabeth Ann Warner Elliott (1935-2009). They were honest, humble, and hard-working
people, but those descriptions don?t begin to describe how important their example was to
me. They blessed my life immeasurably. My brother David was a part of that childhood
as well and I would not be what I am today without him. He continues to be an example
of humility, honesty, and integrity.
vii
Table of Contents
Abstract ....................................................................................................................................... ii
Acknowledgments........................................................................................................................ iv
List of Figures ........................................................................................................................... xiv
List of Tables .............................................................................................................................. xv
Chapter 1: Introduction ............................................................................................................... 1
Statement of the problem ................................................................................................. 1
Theoretical foundations in representation ........................................................................ 3
Purpose of the study .......................................................................................................... 4
Significance of the study ................................................................................................... 5
A brief summary of the content to follow ......................................................................... 6
Chapter 2: Review of the Literature ............................................................................................ 7
Technology in mathematics education .............................................................................. 8
Benefits of technology to mathematics education ................................................ 9
Concerns regarding technology in mathematics education ................................ 11
Adult developmental mathematics students ................................................................... 14
Introduction to adult developmental mathematics education ............................. 14
Challenges faced by adult developmental mathematics students ....................... 19
Technology use in adult developmental mathematics education ........................ 23
Issues in technology use for adult students ............................................. 24
viii
Computer assisted instruction ................................................................. 25
Helping instructors make technology choices ........................................ 34
Representation in mathematics education ....................................................................... 35
Representational systems within which learners operate .................................... 40
Representations unique to the learner ................................................................. 44
The role of visualization and imagery ................................................................ 46
Symbolization as a construct related to representation ....................................... 47
Connecting multiple representations ................................................................... 50
Uses of representations: Models and functions .................................................. 53
Models as representations ....................................................................... 53
Functions as a context for studying representation ................................. 55
Building validity, usefulness, and endurance ..................................................... 58
Valid internal representations ................................................................. 59
Useful internal representations ............................................................... 60
Enduring internal representations ........................................................... 61
The connections between technology and representation ............................................... 62
Recent studies in technology and representation ............................................................ 71
Using relations to look at algebra from a geometric perspective ........................ 71
The use of spreadsheets as cognitive tools ......................................................... 71
The use of object oriented programming to build self-efficacy .......................... 72
Exploring connections with graphing calculators and dynamic software .......... 74
Using technological laboratories to connect to real-life phenomenon ................ 76
Linked whole group study .................................................................................. 79
ix
The use of dynamic geometry environments to deepen thinking ....................... 80
Theoretical framework: Constructivism ........................................................................ 82
Conclusions and questions ............................................................................................. 84
Chapter 3: Methodology ........................................................................................................... 87
Theoretical foundations for qualitative research ............................................................. 88
Qualitative inquiry and foundations of thought ................................................. 88
Qualitative research in mathematics education ................................................... 90
Specific approach: Teaching experiment ............................................................ 91
Exploring student thinking ...................................................................... 91
Case studies and teaching experiments ................................................... 93
What students do vs. what they might do ................................... 94
Descriptions vs. adjustments ....................................................... 95
Tasks for teaching experiments .............................................................. 98
Approach to data analysis: Grounded theory ..................................................... 100
Concluding ideas about methodological theory ................................................. 102
Methodology learned through the pilot study .................................................... 102
Procedure ...................................................................................................................... 104
Selection of subjects ......................................................................................... 104
Institutional selection ............................................................................ 105
Case selection........................................................................................ 106
Instrumentation and data collection .................................................................. 108
Interview technique ............................................................................... 108
Initial interview ..................................................................................... 109
x
Teaching experiment interviews and tasks ........................................... 109
Technological procedures ................................................................................. 110
Data analysis ..................................................................................................... 112
Coding techniques ................................................................................. 113
Indicative movements ........................................................................... 115
My stance as a researcher ............................................................................................. 116
Reliability and Validity ................................................................................................. 117
Reliability and validity in qualitative research ................................................. 118
Measures taken in the present study ................................................................. 119
Questioning during the course of the study .......................................... 120
Inter-rater reliability .............................................................................. 121
Internal validity ..................................................................................... 123
Talk-aloud protocol ................................................................... 123
Questioning for validity ............................................................ 124
External validity .................................................................................... 124
Conclusions .................................................................................................................. 125
Chapter 4: Results ................................................................................................................... 127
Description of subjects .................................................................................................. 128
The teaching experiment sessions ................................................................................. 131
Marlon?s sessions .............................................................................................. 134
Marjorie?s sessions............................................................................................ 138
Major themes arising from the study ............................................................................ 139
Mathematical thinking processes ...................................................................... 141
xi
Algebraic misconceptions ..................................................................... 142
Function and coordinate point confusion .............................................. 144
Graphical confusion .............................................................................. 145
Observing patterns, problem solving, sense making and reasoning ..... 146
Summary of mathematical thinking processes ..................................... 150
Representational ideas and issues ..................................................................... 152
Mathematical language ......................................................................... 152
Validity and usefulness ......................................................................... 153
Endurance ............................................................................................. 156
Indicative movements and multiple representations ............................. 157
Internal representations ......................................................................... 160
Summary ............................................................................................... 162
Influences and uses of technology .................................................................... 165
Technology as an aid to mathematical communication ........................ 165
Technology as an aid to reasoning ........................................................ 167
Using technology to reveal and clear up misconceptions ..................... 169
Marlon?s case ............................................................................ 169
Marjorie?s case .......................................................................... 173
Technology as a window into student thinking ..................................... 173
Technology as an aid in the use of standard representations ................ 175
Empowerment through the use of technology ...................................... 177
Examples of empowerment from Marlon?s work ...................... 177
Examples of empowerment from Marjorie?s work .................... 179
xii
Summary .............................................................................................. 182
Chapter summary .......................................................................................................... 184
Chapter 5: Discussion ............................................................................................................. 189
Summary of chapters 1-3 .............................................................................................. 189
Limitations .................................................................................................................... 192
Conclusions ................................................................................................................... 193
What internal representations followed the use of technology? ....................... 194
Connecting multiple representations ..................................................... 195
Building representations based on their own thinking .......................... 197
What was determined about validity and usefulness? ...................................... 198
How well did those representations endure over a period of time? .................. 200
Implications................................................................................................................... 201
Adult developmental mathematics students...................................................... 201
Teachers of adult developmental mathematics students ................................... 203
Design of developmental mathematics programs ............................................. 204
The general use of technology in mathematics education ................................ 206
Further research ................................................................................................ 207
Conclusion .................................................................................................................... 209
References ............................................................................................................................... 212
Appendix A: Glossary of terms ............................................................................................... 228
Appendix B: Interview protocols ............................................................................................ 233
Appendix C: Tasks ................................................................................................................... 243
Appendix D: Coding guide ...................................................................................................... 263
xiii
Appendix E: Consent forms ...................................................................................................... 274
xiv
List of Figures
Figure 1: Diagram of relationships among representational systems ......................................... 41
Figure 2: Diagram summarizing ideas from the literature .......................................................... 85
Figure 3: Still capture illustrating the camera set up and assembled video .............................. 112
Figure 4: Representations Marlon created to show mathematics he remembered .................... 142
Figure 5: Marlon?s work analyzing the pattern in ?Looking at dot patterns? ........................... 148
Figure 6: Marlon?s situation when connecting paper and technological representations ......... 159
Figure 7: Screen shot of Marlon?s graphed points for Another dot pattern .............................. 166
Figure 8: Screen shot of situation at the time Marlon analyzed h(x) = 8 + 9 ........................... 170
Figure 9: Screen shot of situation at the time Marjorie predicted the location of 2x ............... 186
xv
List of Tables
Table 1: Benefits and concerns of the use of technology in mathematics education ................... 8
Table 2: Adult developmental mathematics education summary of cited statistics ................... 15
Table 3: The importance and benefits of adult developmental mathematics education ............. 17
Table 4: Needs of adult learners ................................................................................................. 22
Table 5: Issues in technology use for adult students ................................................................... 25
Table 6: Advantages and disadvantages of computer assisted instruction ................................. 26
Table 7: Contributors to the study of representation in mathematics education ......................... 37
Table 8: Interpretive framework for ideas related to representation ........................................... 59
Table 9: Studies in technology and representation ..................................................................... 66
Table 10: Subjects of teaching experiment ............................................................................... 130
Table 11: Content of teaching experiment sessions .................................................................. 131
Table 12: Data related to mathematical thinking processes ...................................................... 151
Table 13: Data related to representational ideas and issues ...................................................... 163
Table 14: Data related to the influences and uses of technology .............................................. 182
1
1. Introduction
In a junior high school in the southern United States, students have access to laptop
computers and regularly interact with technological representations of mathematics. These may
include charts, graphs, geometric shapes, algebraic equations, or other mathematical objects.
Those objects may be represented via mathematics software, internet sites, or on hand-held
devices such as calculators. Many of the students at this school are accelerated beyond the
normal curriculum for their grade, increasing their future educational opportunities. Across the
river at a mid-sized university, over 500 adult developmental mathematics students try to make
up for lost opportunities to learn. Technology laboratories are provided for them at the
university, but the students may be either reluctant to use them or unaware of their potential. This
situation is symptomatic of issues regarding the use of technology in mathematics education.
New mathematics technology continues to be developed, and may even be made available to
teachers and students, but is it being used wisely to help the students that are most in need of the
help it can provide? Examining the situation of adult developmental mathematics students may
be an important avenue for the examination of this question.
Statement of the Problem
The need for more attention to ways to incorporate technology into developmental
education has been documented (Epper & Baker, 2009; Caverly et al., 2000). Developmental
education, sometimes referred to as remedial education, refers to educational efforts which serve
college students who need additional preparation in order to be successful (Payne & Lyman,
2
1996). Developmental mathematics ?has become an insurmountable barrier for many students,
ending their aspirations for higher education? (Bryk & Treisman, 2010, p. B19). Instructors may
not be familiar enough with the learning barriers that adult students face to choose technology
which will meet their needs. Adult learners also may not have the verbal comprehension
necessary to successfully interact with software designed to supplement instruction (Li &
Edmonds, 2005). The use of software designed to tutor them, provide them with extra practice,
and sometimes engage them in dialog is sometimes known as computer-assisted instruction
(CAI) and it is common in adult developmental mathematics programs ("CAI," 2003; Caverly et
al., 2000). Other uses of technology in developmental programs include internet sites, distance
learning technology, computer algebra systems, graphing calculators and spreadsheets (Epper &
Baker, 2009).
The American Mathematical Association of Two-Year Colleges (AMATYC) has adopted
the use of technology as one of its basic principles (American Mathematical Association of Two-
Year Colleges, 2006; Epper & Baker, 2009). AMATYC?s (2006) document Beyond Crossroads,
which seeks to provide help in implementing mathematics education standards for those teaching
beginning college students, states that ?Technology should be integral to the teaching and
learning of mathematics? (p. 11). The description of the principle states
Technology continues to change the face of mathematics and affect the relative
importance of various concepts and topics of the discipline. Advancements in technology
have changed not only how faculty teach, but also what is taught and when it is taught.
Using some of the many types of technologies can deepen students? learning of
mathematics and prepare them for the workplace (p. 11).
3
Even though it is considered integral to teaching and learning and has the potential to
exert a positive influence, innovation in technology has outpaced its evaluation (Epper & Baker,
2009). There are also many questions still to be resolved in developmental mathematics such as
curriculum content and sequencing which affect the use of technology. Software packages may
follow a broad and shallow curriculum that is inappropriate for the needs of adult developmental
mathematics students. Epper and Baker (2009) have suggested several steps to improve the use
of technology in adult developmental mathematics education including blending best practice
with leading technological innovations, providing greater research evidence, increasing
technological development for education, and overcoming resistance to change in the community
college culture.
Theoretical Foundations in Representation
In order to provide the research evidence which adult developmental mathematics
educators need, it is necessary to understand the use of representation in mathematics education
because of the connections between technology and representation. The National Council of
Teachers of Mathematics (NCTM) (2000) defined mathematical representation as both ?the act
of capturing a mathematical concept or relationship in some form? and ?the form itself? (p. 66).
They also referred to the influence of computers and calculators on representation, noting that
technology has increased the number of representations available to students (NCTM, 2000).
They also noted that technology allows students access to more representations, some of which
students may not otherwise be able to access. New dynamic technology, which allows the
movement of an object represented technologically, often affecting the movement of another
connected object, transforms the possibilities for representation and may have a great impact on
how mathematical objects are conceptualized and mathematical meanings are internalized
4
(Falcade, Laborde, & Mariotti, 2007; Moreno-Armella, Hegedus, & Kaput, 2008). Researchers
have noted the connection between technology and representation and the need for further study
of these connections (Hollenbeck & Fey, 2009; Stylianou, Smith, & Kaput, 2005).
Many studies over the past ten years have looked at links between the use of technology
and representation (cf. Abramovich & Ehrlich, 2007; Falcade, Laborde, & Mariotti, 2007; Kaput,
1998; Yerushalmy & Shternberg, 2001). A variety of technologies in various settings have been
examined, such as a professional development setting in which spreadsheets were examined as
cognitive tools (Alagic & Palenz, 2006); the creation of computer technology to link middle
grades classrooms for a study of multiple representations of functions (Hegedus & Kaput, 2004);
the examination of the use of calculator based laboratories (CBLs, which use sensors to translate
real-world information into calculator data for analysis) with students (Lapp & Cyrus, 2000) and
preservice teachers (Sylianou, Smith, & Kaput, 2005). The idea of a function, central to some of
these studies, is a mathematical relationship in which one set of data is matched with another set
of data so that each piece of data in the input set is matched to one and only one piece of data in
the output set. It is one of the most important topics in mathematics (O'Callaghan, 1998).
Purpose of the Study
This study seeks to add to the work which has been done linking technology and
representation and broaden its scope to specifically address the needs of adult developmental
mathematics students. It has been shown that technology has a potential impact on learners?
conceptualizations and internalization of mathematical meaning (Moreno-Armella, Hegedus, &
Kaput, 2008). It has also been shown that teachers of adult students need more understanding of
their learning barriers (Li & Edmonds, 2005). The focus of this study is on the effect of
technology on adult developmental mathematics students? understanding as evidenced by the
5
apparent changes in their internal representations of mathematics - those mathematical forms
which exist within the students? mind (Goldin, 2003). Those apparent changes and their
relationship to the subject?s interaction with mathematics technology will be examined. The
questions specifically are these:
1. Following the introductory use of dynamic computer technology to explore mathematical
concepts built upon previous knowledge, what internal representations of those concepts
do developmental mathematics students possess?
2. What can be determined about the validity and usefulness of those representations?
3. How well do those representations endure over a period of time and in the company of
tasks which build upon them?
Significance of the Study
The current study has the potential to assist developmental educators in meeting the
needs of adult learners. It can help those educators realize the potential of technology to improve
developmental mathematics instruction for all students (Epper & Baker, 2009). Bryk and
Treisman (2010) recently noted the dilemma of developmental mathematics students who may
work under great pressures in their personal lives. They may put all the effort they can into
completing the mathematics sequences designed to help them get ahead and still fail to complete
them. Describing the case of a single mother working the late shift at a supermarket and trying to
go back to school, Bryk and Treisman (2010) noted that she said ?I just couldn?t do it anymore.?
They noted that for "this student and too many others, the dream stops here" (p. 19). They also
noted that as many as 70 percent of students placed in developmental mathematics courses do
not complete them. Technology has the potential to change such students? lives (Epper & Baker,
2009). The dream need not stop. The results of this study can provide adult developmental
6
mathematics educators with information that will help them to better understand their students?
thinking and make more informed choices as to the technology-based mathematics instruction
they provide those students.
A Brief Summary of the Content to Follow
In chapter 2, the review of the literature will examine the role of technology in
mathematics education, the needs and concerns of adult developmental mathematics students?
mathematical representations, and ideas relating technology and representation as well as
specific studies combining the two. Literature related to the ideas of constructivism will be
presented in order to provide a theoretical setting for the study of students? interactions with
technological representations. Chapter 3 will first discuss the theoretical basis for the methods
chosen and then describe the specific procedures. The theoretical presentation will conclude with
a brief look at lessons learned during the course of a pilot study. In chapter 4, following an
introduction to the two subjects of the case, the progress of the teaching experiment in each of
their cases will be described. This will be followed by a look at the theoretical ideas which were
investigated and discovered as they emerged from the data. In chapter 5 limitations and
conclusions will be presented, including a discussion of what was learned about the research
questions. Implications for teachers of adult developmental mathematics students will also be
noted as will suggestions for further research. It is hoped that these results may help empower
adult developmental mathematics students for a viable academic future.
7
2. Review of the Literature
In order to understand the influence of technology use on the mathematical thinking of
adult developmental mathematics students, it is important to understand issues related to
technology use in mathematics education in general and issues facing adult students. Ideas
surrounding the use of mathematical representations are also important because of their close
connection to the use of technology. This review will begin with an examination of the benefits
and challenges associated with the use of technology in mathematics education. Following this
the particular needs of adult developmental mathematics students will be considered. This will
include information about their general needs as well as the issues related to the use of
technology which they face. The role of representation in mathematics education will be
carefully examined. This examination will conclude with the presentation of an interpretive
framework with which student thinking might be considered. Once this framework has been
established, the connections between technology and representation will be more carefully
considered and several studies relating the two will be examined. Because knowledge and the
development of knowledge is to be examined, an epistemology must exist as part of that
examination, and so literature related to constructivism will also be presented. This will provide
a theoretical foundation for understanding not only student thinking, but the way knowledge of
student thinking can be built. Conclusions and research questions will follow. Note that a
glossary defining key terms is provided in Appendix A.
8
Technology in Mathematics Education
Educational technology continues to be reinvented at a rapid pace, and it is sometimes the
case that research, access, and implementation have difficulty keeping up with the pace of
invention and with each other (Fey, 1984; Fey, Hollenbeck, & Wray, 2010). In order to serve
students, promote positive change, and understand the role of technology in mathematics
education, both the benefits and the challenges associated with it must be considered. Table 1 is
provided below as a summary of some of these ideas.
Table 1
Benefits and Concerns of the use of Technology in Mathematics Education
Benefits
Concerns
New forms of mathematical activity (Moreno-
Armella et al., 2008)
Opportunity for student initiative and
exploration (Fey, 1984; Fey et al., 2010)
The ability to visualize as mathematicians do
(Cuoco & Goldenberg, 1996)
Provides external reference objects which
helps make their thinking explicit and clarify
their ideas (Hennessy, Fung, & Scanlon, 2001)
Quicker problem solving ?feedback loops?
(Shaffer & Kaput, 1998, p. 111)
May serve the role of active listener (Connell,
1998)
Ensuring that conceptual understanding is
center stage and student thinking is not
replaced (Fennell & Rowan, 2001; Fey et
al., 2010)
Both technical and mathematical knowledge
are needed to take advantage of learning
possibilities (Lingefjard, 2008)
Selection of appropriate tasks is important
to taking full advantage of technology use
(Alagic & Palenz, 2006; Gadanidis et al.,
2004)
9
Benefits
Concerns
More direct access to mathematics structures
allowing a greater impact on their minds
(Moreno-Armella et al., 2008)
Encourages dynamic visualization
(Yerushalmy et al. 1999, cited by Presmeg,
2006)
Encourages dynamic visualization
(Yerushalmy et al. 1999, cited by Presmeg,
2006)
Move more flexibly between different
representations (NCTM, 2000)
Time, training, and immersion in the
technology are needed for those using it to
learn to think with technology rather than
about it (Gadanidis, 2008)
Gaps exist between the level of technology
available and the practical use being made
of it (Atan, Suncheleev, Shitan, & Mustafa,
2008; Hollenbeck & Fey, 2009; Oncu,
Delialioglu, & Brown, 2008)
Benefits of technology to mathematics education. Digital technologies are capable of
providing new forms of mathematical activity in socially rich ways (Moreno-Armella et al.,
2008). They redefine the practices, content, and ways of knowing about a subject. If a certain
technology is absent, different knowledge will be produced (Villarreal, 2008). When present,
technology can provide an opportunity in the learning environment for student initiative and
exploration (Fey, 1984; Fey et al., 2010).
Mathematicians have been known to use creative imagery and metaphor to understand
and think about mathematics. Some researchers believe that through technology, students can be
10
provided with the opportunity to "tinker with mathematical objects just as they might tinker with
mechanical objects" and thus develop the ability to visualize as mathematicians do (Cuoco &
Goldenberg, 1996, p. 17). Immersion in dynamic, technological mathematical environments
could have a great impact on how students conceptualize mathematical objects and what they
consider doing mathematics to entail (Moreno-Armella et al., 2008). Dynamic environments
allow mathematical representations to be set in motion in some way. Not all technological
representations of mathematics are designed to do so, as is the case with some types of computer
aided instruction (CAI) software which may only involve the static input of student responses to
questions (Kaput, 1992). The influence of dynamic computer environments continues to be an
area of significant focus in the study of education for effective mathematical visualization
(Presmeg, 2006). Researchers have conjectured that computer models can provide students the
start they need to be able to engage in more advanced mental visualization of mathematical
concepts (Cuoco & Goldenberg, 1996).
Technology can provide an external reference object which encourages students to make
their thinking explicit and clarify their ideas (Hennessy, Fung, & Scanlon, 2001). It allows
students greater opportunity to experience mathematics as an "experimental enterprise.? The
problem solving "feedback loops" are quicker and not dependent on symbolic manipulation
(Shaffer & Kaput, 1998, p. 111). Dynamic media allow students more direct access to
mathematical structures and thus allow those structures to have greater impact on the minds of
students (Moreno-Armella et al., 2008). Yerushalmyet al. (1999, as cited in Presmeg, 2006)
noted that the use of computer software for mathematics encourages dynamic visualization. The
visual process used to classify the types of problems in their study was enhanced by the use of
dynamic software which allowed the users to move flexibly between different representations.
11
Iterative examples developing the concept of limit and the asymptotic behavior of certain
functions, and transformations are some of the mathematical topics made more accessible
through the use of technology (NCTM, 2000).
Concerns regarding technology in mathematics education. Care must be taken to
ensure that conceptual understanding, rich in connections to other ideas, is center stage and that
students have the chance to produce their own representations of what is occurring within the
technological representation (Fennell & Rowan, 2001; Goldin, 2003; Lapp & John, 2009).
Sometimes materials used to represent mathematical thinking replace student?s thinking rather
than represent it (Fennell & Rowan, 2001). Good representations show how students are
thinking. Students may be taught how to use tools such as hand-held modeling objects known as
manipulatives or technological representations as the only way to solve problems and when this
happens, the tool may actually interfere with learning and fail to build mathematical
understanding. Forms of representations can become ends in themselves, which is not productive
for students. (Fennell & Rowan, 2001). It is just as harmful to use manipulatives or technological
tools in a formulaic "do as I do" way as it is to have students blindly follow algorithms. The goal
is to have students use manipulatives and technological tools to think with rather than as answer
machines. Van de Walle (2007) stated that "A mindless procedure with a good manipulative is
still just a mindless procedure" (Van de Walle, 2007, p. 34). When the focus of teaching is on
attitudes, atmosphere, and objectives, and the teaching materials are seen as a means to
maximize those aspects of the lesson, then those materials are not an end in themselves
(Villarreal, 2008). Rather than being used for mindless procedures, the computer may, for
example, serve the role of an active listener, doing as the student tells it to do, and assisting the
student in constructing their own knowledge (Connell, 1998).
12
Students must have both technical and mathematical knowledge to take advantage of the
possibilities for learning that a technological tool provides (Lingefjard, 2008). This requires time
and training if students are to learn to think with the technology. When human beings immerse
themselves in using a technology, then they can learn to think with that technology, rather than
about that technology (Gadanidis, 2008). Selection of problems is another important aspect of
accomplishing learning goals in a technology oriented environment. This is particularly
important when time must be taken to assist students in becoming proficient with the tools being
used (Alagic & Palenz, 2006).
Interactive examples may be accompanied by instructions for students and teachers that
have been designed to foster investigation, problem solving, and student discovery. Those
examples, however, may not support the discovery envisioned by their creators (Gadanidis,
Sedig, Liang, & Ning, 2004). Such tasks should be analyzed to see whether or not they support
the desired pedagogy. One quality to look for is whether or not the task provides appropriate
mathematical patterns related to the investigation, so that students may observe the relationships
found in the objectives for the lesson. For example, if the investigation calls for students to
analyze relationships between surface area and volume, are those outputs displayed together on
the same graph or separately on different graphs? What would help the student more? Are
different helpful representations present, such as equations, ratios, tables, graphs, and
illustrations, if they would prove useful to the student? (Gadanidis et al., 2004). Even considering
the benefits and attractions of interactiveness, students may be more likely to engage with an
investigation involving real-life content than a purely mathematical investigation (Gadanidis et
al., 2004). Those producing interactive visualizations for the classroom would benefit from using
a review process including feedback from classroom teachers and suggestions for improvement.
13
The design of such tasks should include consideration of principles of both presentation and
interaction, incorporating the maximum mathematical benefit for the student. This includes
making the information intelligible, engaging the learners with the information and facilitating
mental interaction with the material (Gadanidis et al., 2004).
Though technology has advanced, the use of technology has not always kept pace with its
development. Access to technological tools is easier than the far greater challenge of determining
how to use those tools effectively (Fey et al., 2010). There may be gaps between the level of
technology available and its practical applications in mathematics education (Atan, Suncheleev,
Shitan, & Mustafa, 2008; Hollenbeck & Fey, 2009). Merely using the technological tool to find
solutions does not guarantee knowledge of broader principles. Care must be taken that student
conjectures arising from the use of software are subject to appropriate mathematical proof (Fey
et al., 2010). It should also be noted that despite the many advantages of technology, paper and
pencil provides a recording media which leaves an accessible record of what happened,
something which technology cannot always provide (Goldin, 2003).
Fey spoke in 1984 of the challenges of incorporating technology into mathematics
education. He noted then that the traditional pattern for educational change, in which a proposal
is made by professional educators, curriculum is written, and classroom implementation follows,
underestimates the complexity of the actual process of change (Fey, 1984). In some instances,
technology may be available to teachers, but may not be used to support student learning (Oncu,
Delialioglu, & Brown, 2008). Hollenbeck and Fey (2009) illustrated how a variety of the latest
technological tools, if available to teachers, could be used to enhance learning in the mathematics
classroom. They concluded that appropriate use of technological instructional tools should be a
high priority for mathematics education researchers (Hollenbeck & Fey, 2009). This echoes the
14
suggestion Fey made in 1984 that, in spite of the challenges in implementation, the best use of
technology in the classroom be determined without the research being limited by concerns over
how to make that technology available. One of the populations in need of such work is adult
learners. Following is an examination of their general needs and then a more specific look at the
role of technology in their mathematics education.
Adult Developmental Mathematics Students
I will first present some general information about adult developmental mathematics
education. I will then look at particular challenges faced by the students in these programs.
Following this, will be an examination of the use of technology for adult developmental
mathematics students.
Introduction to adult developmental mathematics education. One of the subjects of
the current study instructed the tutors in the mathematics lab she frequented to ?pretend? she was
?in middle school.? She seemed to understand and reflect the truth that many students in the U.S.
begin to fall behind at the point in their education where algebra course work customarily begins
(Epper & Baker, 2009). This is the mathematical content on which developmental students are
assessed for placement. Developmental mathematics programs may be the key to success for
many students. They seek to bridge the gap between what has been learned in high school and
what is needed for success in postsecondary education (Epper & Baker, 2009). Without
developmental education, many people would never have the opportunity to attend college or
improve their employment possibilities (Gerlaugh, Thompson, Boylan, & Davis, 2007). Table 2
displays some of the statistics regarding adult developmental mathematics education. Title IV
institutions are those which participate in certain federal student aid programs (Aud et al., 2010).
15
Table 2
Adult Developmental Mathematics Education Summary of Cited Statistics
Item
Percent Source and year
All Title IV institutions offering developmental mathematics 71% Parsad and Lewis
(2003) NCES report
Public 2-year institutions offering developmental mathematics 97% Parsad and Lewis
(2003) NCES report
Public 4-year institutions offering developmental mathematics 78% Parsad and Lewis
(2003) NCES report
Freshmen surveyed at all types of institutions who were enrolled
in developmental mathematics
>20% Parsad and Lewis
(2003) NCES report
Private 2?year institutions freshmen surveyed who were
enrolled in developmental mathematics
33% Parsad and Lewis
(2003) NCES report
Students in 2-year colleges requiring developmental
mathematics
>50% Shwarte (2007)
Those who test into developmental mathematics that pass on the
first attempt (some states)
40-
50%
Trenholm (2006)
Percent of students in Nevada graduating from high school in
2006 and attending college the following fall that enrolled in
remedial mathematics during their first year of college
37.6% Fong, Huang, &
Goel (2008)
NCES report
More developmental courses in mathematics are generally offered than in the other two
standard developmental topics, reading and writing (Parsad & Lewis, 2003). The National Center
for Educational Statistics (NCES) (Parsad & Lewis, 2003) published a special report in 2003
16
focusing on what they call remedial education. They reported that in the year 2000, 71% of all
Title IV degree-granting institutions with freshmen offered developmental courses in
mathematics. When broken down further, it was noted that 97% of public 2-year institutions
offered developmental courses in mathematics, as did 78% of public 4-year institutions. More
than one fifth of freshmen at all institutions in the survey enrolled in developmental mathematics
as freshmen. For private 2-year institutions, this figure rises to 35% (Parsad & Lewis, 2003). In a
later study published by NCES, Fong, Huang, and Goel (2008) reported that of 4,653 students
graduating from Nevada public high schools in 2006 and enrolling in at least one mathematic
course in a Nevada public college the following school year, 37.6% enrolled in a remedial
mathematics course. Schwarte (2007) stated an even higher figure, noting that more than half of
the students in two-year colleges require developmental mathematics instruction to prepare them
for college-level mathematics work. Trenholm (2006) noted further that in some states, only 40-
50% of those that test into developmental mathematics courses pass on the first attempt. These
figures show the large population being served, the need for improvement, and the implication
that research into adult developmental mathematics education is important.
Lesik (2007) showed that developmental mathematics can be effective, noting that the
risk of college drop-out among developmental mathematics students was significantly lower than
for similar students who did not participate in developmental mathematics programs (Lesik,
2007). Her study followed 1,276 freshmen at a 4-year institution and noted that 536 eventually
dropped out. She reduced the sample to those students who were similar, that is who scored
"within 5 points on both sides of the cutoff score" on a placement test which determined whether
or not they were assigned to developmental mathematics courses. This resulted in n = 212 (p.
597). A statistical analysis showed that the risk of a student dropping out was significantly lower
17
for similar students who did participate in developmental mathematics than it was for those who
didn't participate in developmental mathematics. After the first year, those who did participate
had "an estimated risk of dropout" of 8.2% (p. 601). For those who did not participate, the risk
was 27.7%.
A study by Duranczyk and Higbee (2006) showed that students and colleges both benefit
from developmental mathematics programs at 4-year institutions, as well as at 2 year institutions.
They surveyed 20 and interviewed 18 people who had completed developmental coursework at
"a comprehensive, urban, public university in the Midwest" (p. 24). One practical aspect of being
required to take developmental courses at a different location is the inconvenience of working
through the registration process of two different institutions. Some respondents also indicated
that the convenience of being able to easily transfer from a course in which they were enrolled,
but having difficulty, to one which would bolster their chances of success at the same institution
was a factor in their eventual success (Duranczyk & Higbee, 2006). Table 3 summarizes some of
these ideas related to the importance and benefits of adult developmental mathematics education.
Table 3
The importance and benefits of adult developmental mathematics education
Author
Year Major ideas
Gerlaugh, Thompson,
Boylan, & Davis
2007 The importance of
developmental education in
opening the door to
opportunity
18
Author
Year Major ideas
Lesik 2007 Drop-out rates among
students enrolled in
developmental mathematics
lower than for similar
students who were not so
enrolled
Duranczyk and Higbee 2006 Benefits of having
developmental mathematics
available at 4-year
institutions shown
Galbraith and Jones 2008 Describe challenges of
teaching adults
Interested parties, including public policy groups and researchers, who have noted the
role of mathematics as a gatekeeper to future success, have focused attention on developmental
mathematics education and the feasibility and desirability of new strategies and innovations
(Epper & Baker, 2009). Regarding those who teach developmental mathematics students,
Galbraith and Jones (2008) have noted, "Teaching developmental mathematics in a community
college is demanding, challenging, and rewarding for those who engage in the endeavor" (p. 35).
Despite the effectiveness of developmental mathematics in many cases, the numbers cited by
Trenholm (2006) are unacceptably high. Alternative educational methods for this population
19
must be considered (Trenholm, 2006). In order to assess such methods, it?s necessary to
understand the needs of adult learners. A look at some of the challenges they face follows.
Challenges faced by adult developmental mathematics students. Special needs of
adult developmental mathematics students include understanding and navigating the educational
system, accurately assessing their own needs, managing non-cognitive factors affecting their
education, and access to knowledgeable teachers and other resources. Non-cognitive factors
include influences unrelated to the student?s knowledge. They include factors such as self-
efficacy and motivation, which have been shown by research to be important in developmental
mathematics achievement. In their study, Wadsworth, Husman, Duggan, and Pennington (2007)
defined self-efficacy as a person?s belief in their own ability to be effective in managing future
situations. They looked at achievement along with measures of learning strategies and self-
efficacy in 89 developmental mathematics students enrolled in an online course. They measured
self-efficacy by asking students to rate their confidence in their ability to complete certain types
of mathematics problems relevant to the course in which they were enrolled. They found that
achievement was in part affected by self-efficacy and certain learning strategies (motivation,
concentration, information processing, and self-testing) (Wadsworth et al., 2007). Obiekwe
(2000) in his discussion of the instrument used by Wadsworth et al. noted that concentration was
thought of as a student?s ability to give attention to an academic task. He described self-testing as
a student?s ability to prepare for tests and classes, and information processing as a student?s
ability to process knowledge. Wadsworth et al., (2007) noted that direct instruction, a term they
seem to use to indicate in-classroom instruction with a teacher present, as a supplement to
computer instruction can help students improve in these areas. For example, online course
20
instructors can meet face to face with students and discuss with those students the learning
strategies they will need in order to be successful.
In addition to low self-efficacy and a need for better learning strategies, research has
identified some of the other factors that affect adult students. For example, those who are placed
in remedial mathematics courses are disproportionately minority and first-generation college
students, placing them at risk (Epper & Baker, 2009). Some have also found that a higher
percentage of developmental mathematics students have learning disabilities (Epper & Baker,
2009). First-generation college students, recent immigrants, and students of color tend to be at a
disadvantage because of their family's lack of understanding of the educational system and the
implications for the future of non-college high school tracking (Collins, Bollman, Eaton, Otte, &
Thomas, 2000). Those whose education includes a significant time gap between the completion
of their last mathematics course in high school and the beginning of their first college course
may find it difficult to be successful in their college mathematics course (Collins et al., 2000).
Some who struggle in college mathematics may have difficulty correctly identifying what it is
that is keeping them from succeeding (Hall & Ponton, 2005)
Standardized tests, state placement tests, institutional placement tests, and the student?s
history of courses taken and grades earned all may be factors in students? college mathematics
placements (Collins et al., 2000). "Students may resist or feel insulted by? those placements
(Collins et al., 2000, p. 37). In addition, the placement tests they may be required to take may not
only be far removed from their last mathematics course, as has been noted, but also may not
match the instruction they were given. For example a student who took a calculator based
curriculum in high school may not be permitted the use of a calculator on the placement exam
(Collins et al., 2000).
21
This discussion shows that adult developmental mathematics students may need
assistance outside of class to address both academic and non-academic factors. Research
indicates that non-cognitive factors, such as time management and motivation, influence
developmental students' success, but non-cognitive assessment is infrequently employed
(Gerlaugh et al., 2007). The time demands on adult developmental mathematics students may
make it difficult for them to participate in enrichment programs designed to teach skills needed
for success in college (Collins et al., 2000). Many adult developmental mathematics students
have multiple responsibilities outside of school and value flexibility of delivery and readily
available support services (Epper & Baker, 2009).
One of the challenges developmental education faces is that faculty have come from
other fields and have not had professional training in dealing with developmental students. They
may also not have had training in teaching with technology (Caverly et al., 2000). Though
developmental mathematics instructors may all believe that mathematical understanding is their
primary goal, they do not all think of that understanding in the same way. Many believe
mathematical understanding to be procedural, consisting of an ability to perform a sequence of
actions. Others believe it to imply conceptual knowledge, which has been described as
knowledge that is part of a network of connections (Kinney & Kinney, 2002). Developmental
mathematics teaching practices currently emphasize procedural fluency over conceptual
understanding (Epper & Baker, 2009). Faculty members may not have had enough professional
development opportunities. They are also under pressure to quickly do the job that was not done
in high school, and may feel there is not time to teach for both concept and fluency. Some argue
that teaching conceptually will aid in fluency. Those who seek to make improvements in courses
may, nevertheless, reduce content or increase the number of courses in which the content will be
22
covered. Some believe there is not enough time to teach what is required without the use of
technology (Epper & Baker, 2009). Table 4 summarizes some of the needs of adult learners
which have been discussed.
Table 4
Needs of Adult Learners
Author Year Key ideas
Wadsworthet
al.
2007 Adult learners achievement affected by self-efficacy and learning
strategies.
Epper &
Baker
2009 Adult developmental populations are disproportionately minority and
first-generation college students. They also have a higher percentage of
learning disabilities as well as many responsibilities outside of school.
Faculty members are under pressure to quickly do a job that was not
done in high school
Collinset al. 2000 Family?s lack of understanding of educational system has affected many
adult developmental mathematics students.
A time gap between high school and college may hinder their success.
?Students may resist or feel insulted by? placement tests (p. 37)
Placement tests may not match the instruction that they received
Hall &
Ponton
2005 They may have difficult identifying what it is that is keeping them from
succeeding
23
Author Year Key ideas
Gerlaugh et
al.
2007 Non-cognitive factors such as time management are important but
seldom assessed
Caverly et al. 2000 Faculty may have come from other fields and lack appropriate
professional development
Technology use in adult developmental mathematics education. Technology found in
use in developmental mathematics classrooms includes computer assisted instruction (CAI)
software, internet sites, distance learning technology, computer algebra systems, graphing
calculators, and spreadsheets (Epper & Baker, 2009). Almost one third of all institutions
indicated that computers are frequently used by students as instructional tools in developmental
mathematics (Parsad & Lewis, 2003). Such technology can be beneficial for developmental
mathematics students.
The American Mathematical Association of Two-Year Colleges (AMATYC) has adopted
the use of technology as one of its basic principles (AMATYC, 2006). Technology, however, can
only truly transform developmental education when it is used to foster change in student
behavior, so that students take control of their own learning and persist toward the successful
accomplishment of their worthy goals (Brothen, 1998).
The following discussion will start with a look at issues particular to adult students using
technology. As very few studies exist which pair adult learners specifically with the use of
dynamic geometry or algebra software, I will present the reader with a closer look at studies
involving CAI, which is the most dominant form of computer technology used with adult
24
learners. The technology choices adult developmental mathematics instructors must make will
then be examined.
Issues in technology use for adult students. Although it has been shown that the use of
technology is critical to the success of developmental mathematics education, innovation has
outpaced evaluation (Epper & Baker, 2009). There are also other challenges to implementation.
Those working with developmental mathematics students must carefully consider the role of
race/ethnicity and prior academic performance, both of which may have bearing on the choices
those students make about the use of educational resources made available to them, including
computer technology (Duranczyk, Goff, & Opitz, 2006).
One factor to consider when planning for the use of technology is the relatively low rate
of computer ownership of students enrolled in developmental mathematics. Computer
laboratories must be available and provide the features needed (Epper & Baker, 2009). For
students to take advantage of the opportunities technology provides, "it must become a seamless
part of the learning environment" (Epper & Baker, 2009, p. 9). Online learning faces the
problems of student discipline, cost, and faculty acceptance (Epper & Baker, 2009).
Instructors' perspectives may influence their decisions as to the use of technology in their
classrooms (Kinney & Kinney, 2002). Epper and Baker (2009) reported tension between
procedural fluency and conceptual understanding approaches in their review of practices in
developmental mathematics education. This tension had implications for the use of technology
(Epper & Baker, 2009). Some have noted that both fluency and conceptual understanding are
vital, but curriculum content and sequencing questions have yet to be resolved. Many
developmental mathematics texts and software packages possess the "mile wide and an inch deep
. . . laundry list" quality that afflicts the U.S. K-12 curriculum (Epper & Baker, 2009, p. 5). More
25
technological applications in developmental mathematics focus on procedural fluency than on
conceptual understanding due in part to the current demands of the market (Epper & Baker,
2009). Table 5 summarizes some of the issues in technology use for adult students.
Table 5
Issues in technology use for adult students
Author Year Concerns noted
Epper & Baker 2009 Low rate of computer ownership among students
enrolled in developmental mathematics
Online learning hampered by student discipline, cost, and
faculty acceptance
Tensions between faculty beliefs regarding procedural
fluency vs. conceptual understanding affect the use of
technology
Software packages may have the same defects that
textbooks have, such as a ?mile wide inch deep?
curriculum or a focus on procedural over conceptual
understanding
Duranczyk, Goff, &
Opitz
2006 Race/ethnicity and prior academic performance may
have a bearing on choices students make about their use
of educational resources available, including technology
Kinney & Kinney 2002 Faculty perceptions affect decisions to use technology
Computer assisted instruction. Computer assisted instruction (CAI), which provides a
tutoring supplement to or in some cases replaces classroom instruction, is found by some studies
26
to be used in more that 40% of community colleges in the U.S. and is frequently referred to in
developmental mathematics education literature (Epper & Baker, 2009). Some have referred to
such computer use as computer-mediated learning, defining it to be learner centered computer
intervention in which the computer provides the instruction, requires student responses, provides
immediate feedback, and tracks students' progress (Kinney & Kinney, 2002). This review will
consider computer assisted instruction, computer-mediated learning as well as online instruction
to all be under the umbrella of CAI. An examination of the issues related to this particular form
of technological intervention may assist in the effective implementation of other forms of
technology as well. Since CAI is the predominant form which technology use in adult
developmental education currently takes, an examination of its use may also point to reasons
why other forms of technology use with adult developmental mathematics students, such as the
one examined in the current study, should also be considered. Table 6 summarizes some of the
advantages and disadvantages to the use of CAI which will be discussed.
Table 6
Advantages and disadvantages of computer assisted instruction (CAI)
Authors Type of work Advantages to the use
of CAI noted
Disadvantages to the use
of CAI noted
Caverly et
al., 2000
Conference paper
summarizing issues
in technology and
developmental
education
Allows students to
move on when they are
ready to do so
May provide only
superficial knowledge
which cannot be applied
in other situations
27
Authors Type of work Advantages to the use
of CAI noted
Disadvantages to the use
of CAI noted
Epper &
Baker,
2009
Overview
summarizing the
issues involved in
using technology to
remediate adult
mathematics
students
Some improved results
Allows an alternative to
regular class meetings
Some reduction in
discrimination
Certain types of CAI
software may
emphasize meaning
Kinney &
Kinney,
2002
Surveyed 11
instructors who had
taught both
developmental level
mathematics
courses using CAI
and those which did
not
Students control the
pace of learning,
receive more
instruction,
receive immediate
feedback with detailed
explanations,
move more quickly,
get more practice, and
remain active during
instructional time
Lack of discussion
Only one way of thinking
presented
Students fail to ask for
help when needed
Students can?t hear
conversations with others
Instructors had difficulty
determining how deeply
students were thinking
28
Authors Type of work Advantages to the use
of CAI noted
Disadvantages to the use
of CAI noted
Li &
Edmonds,
2005
Three basic
mathematics
classrooms of at-
risk adult learners
were compared,
two of which used
CAI and one which
was traditionally
taught. Students
were tested for
knowledge and
attitude.
Increased confidence
level
Increased satisfaction
level
Improved ability to
transfer skills to
classroom settings
Bridges gaps in
classroom instruction
Helps meet needs of
diverse learners
Low literacy skills of
developmental students
hinder their ability to
make use of CAI
Qi &
Polianskaia
2007
Examined
enrollment,
completion, and
assessment data for
traditional and CAI
courses
Those in CAI courses did
no better than their peers
in traditional courses
29
Authors Type of work Advantages to the use
of CAI noted
Disadvantages to the use
of CAI noted
Taylor,
2008
Surveyed freshman
enrolled in
intermediate
algebra courses,
both traditional and
CAI based (online
delivery) to
determine both
attitude and
achievement
Can decrease anxiety in
some cases without
lowering performance
Trenholm,
2007
Surveyed online
mathematics course
faculty members to
determine
assessment
practices
Questions as to proper
assessment must be
addressed when online
delivery is used. 78% of
online developmental
instructors used proctored
exams.
Some studies do show improved results for those who use CAI (Epper & Baker, 2009).
Research has shown that CAI increases confidence levels and satisfaction, improves student
ability to transfer skills learned online to classroom settings, bridges gaps in classroom
30
instruction, and helps meet the needs of diverse learners (Li & Edmonds, 2005). Teachers in one
study felt that the uses of CAI allowed students to control the pace of their learning, to choose
the path that best met their needs, to receive more instruction through multi-media, to receive
immediate feed back on their work with detailed explanations, to move more quickly and get
more practice, and to remain active during their instructional time, rather than passive (Kinney &
Kinney, 2002). It can also be advantageous for adult learners because technology which allows
developmental mathematics students to find an alternative to regular class meetings allows them
to manage their many responsibilities (Epper & Baker, 2009).
Kinney and Kinney (2002) surveyed 11 instructors with experience leading two types of
courses, those using CAI and traditional lecture style courses. The topics they taught ranged from
elementary algebra to college algebra. Though the results showed a slight preference for CAI
courses, they also found some disadvantages to the use of CAI mentioned. Those disadvantages
included lack of discussion, the presentation of only one way of thinking, student failure to ask
for help when they needed it, lack of opportunity for students to hear instructors? conversations
with others about topics they need to understand better, and the inability of the instructor to
know how deeply the students were thinking (Kinney & Kinney, 2002). Some respondents
wanted students to construct more of their own knowledge and gain greater conceptual
understanding. They addressed this issue by incorporating writing into the class, such as daily
checkpoint questions which briefly ask student to clarify a concept or justify a task, and learning
logs, which require more in depth writing than a checkpoint question (Kinney & Kinney, 2002).
Some CAI may be presented through online course delivery. Taylor (2008) conducted a
study in which she surveyed 54 freshman students enrolled in intermediate algebra courses using
computer software and 39 students enrolled in traditional intermediate algebra courses. She
31
administered pre-tests and post-tests for both mathematics achievement and mathematics anxiety
ratings. Results showed that the use of web-based technology can in some cases decrease anxiety
in developmental mathematics students without lowering performance. Some have reported that
online delivery of instruction reduces discrimination (Epper & Baker, 2009). The visual image of
the student in such cases is not a factor, and students who are reluctant to speak up in class may
be less so in an online environment.
Although there are benefits to online delivery, there are also disadvantages. A study by
Trenholm (2007) addressed the question of how learning outcomes for online mathematics
courses could be effectively assessed. It considered what percentage of such courses are
proctored, what the differences might be in proctored and unproctored courses, and what faculty
of online courses considered important to their assessment practices. Three survey questions
were sent to about 120 online mathematics course faculty members. The total final response rate
was 39%. Six courses and ten categories of assessment were considered in the analysis of the
data. Results showed that half of the two-year institutions that responded used proctoring, but
none of the four-year institutes used proctoring. There were significant differences in the
proctoring percentage by course, with a large percentage of developmental courses using
proctoring (78%), more than half of calculus course using proctoring, but only 39% of General
Algebra courses and only 8% of general liberal arts courses. Data is also given about proctoring
for different types of assessments. Unproctored (100% online) courses tended to rely more
heavily on formative assessment, that is, assessment designed primarily to provide constructive
feedback to the student so that he or she may improve. The author concluded that at "this time, in
math e-learning, it appears only some form of significant proctored summative assessment
instrument will ensure that educational standards and integrity are preserved" (p. 53).
32
Other issues that have arisen with the use of CAI involved literacy and experience. Li and
Edmonds noted that "[a]t-risk students with low literacy skills are hindered by their inability to
comprehend written language . . . in a CAI environment" (Li & Edmonds, 2005, p. 162). In
addition, if teachers have not experienced CAI, they have no experiential basis upon which to
decide about its effectiveness (Kinney & Kinney, 2002). The capabilities of the software used,
the physical resources available, the method of implementation, the amount and type of
teacher/student contact, and the teacher?s theoretical beliefs have all affected the implementation
of CAI and could reasonably be expected to affect the implementation of other types of
technology as well.
Teachers who wish to make effective use of CAI may provide visually appealing web
materials, and be readily available to assist students. Teacher should ensure that the format is
easy to navigate. They would also better serve students by helping them connect what they are
learning to their life's goals (Li & Edmonds, 2005). Instructors incorporating CAI into their
courses may decide not to lecture, when the software used provides what they feel are complete
initial presentations of the material. They may instead provide clarifications, assistance,
feedback, and study skills training. Those who do feel the need to lecture typically use direct
instruction, supplemented by whole class discussions, and opportunities for students to practice
individually or collaboratively (Kinney & Kinney, 2002).
Some schools have redesigned their programs with the help of CAI so that students only
take the portions of the course in which they need remediation. The course is divided into
modules and the modules are combined with CAI, giving different students different software
assignments. Classroom instruction is focused on conceptual understanding and study skills
(Epper & Baker, 2009). Different CAI programs have different capabilities. Cognitive Tutor is a
33
CAI program which emphasizes meaning and fluency, multiple representations, and formative
assessment (Epper & Baker, 2009). Cognitive Tutor was developed by Carnegie Learning and is
described as a research-based product incorporating the opportunity for students to work with
multiple representations and to view examples which are intended to help build conceptual
understanding ("Carnegie Learning," 2010). Epper and Baker (2009) reported that Pellissippi
State Community College in Knoxville, Tennessee redesigned their developmental mathematics
program to include the use of Cognitive Tutor in combination with classroom instruction,
resulting in increased success over traditional teaching methods.
Programs that use CAI are common to developmental education. They may place
students at a certain level, and allow them to move on when a test indicates that they are ready to
do so (Caverly et al., 2000). Such programs have some advantages, however, the behaviorist
model such programs typically follow only provides students with superficial levels of
knowledge. Behaviorist models may provide a stimulus and response approach without regard to
conceptual understanding. Students may be able to recall information, but not be able to apply it.
When technology is used as a tool in a social constructivist setting, the student has a greater
chance of reaching more complex levels of understanding (Caverly et al., 2000).
Qi and Polianskaia (2007) showed that the use of CAI does not necessarily increase
performance. They examined enrollment, completion, and assessment data for traditional and
computer-mediated course at a community college with a population of about 4,000 students.
The computer-mediation was a self-paced multi-media environment called PLATO interactive
mathematics (Plato Learning, 2004). Interactive conceptual presentation, immediate feedback,
skills development, and online quizzes were all part of the software. It also provided teacher
tools such as tracking for student progress and time on task. After a carefulness analysis of the
34
data which compared completion rates, pass rates, and average scores for traditional and
computer-mediated courses, the researchers found that those in the computer-mediated courses
did no better than their peers in traditional courses (Qi & Polianskaia, 2007). Other choices aside
from CAI should be available for developmental mathematics teaching and learning.
Helping instructors make technology choices. The issues surrounding the use of CAI
illustrate the importance of the manner in which technology is incorporated into developmental
mathematics courses. The use of technology in and of itself does not guarantee improvement in
student performance (Qi & Polianskaia, 2007). Technology as it is being used may only be
fostering superficial knowledge (Caverly et al., 2000). Attention to students? learning barriers is
another important consideration developmental educators face (Caverly et al., 2000). Decisions
as to what technology to use and how to use it must be carefully considered in order to best meet
the needs of students. Educators must have the knowledge they need to make these choices. They
must be familiar with the way each particular type of technology affects student thinking and
learning, since the capabilities of software is one of the factors in its implementation. Some CAI
use, for example, may hamper valuable mathematical communication and instructors may find it
challenging to determine how deeply students are thinking (Kinney & Kinney, 2002). On the
other hand, it has been shown that the use of dynamic interactive technology can, in some
students, foster new understanding of mathematical concepts with which they have previously
struggled (Li & Edmonds, 2005). It has also been shown that mathematical software which
allows constructive explorations can help build higher levels of understanding. Allowing
students to use technology which lets them create tools for other students strengthens this
knowledge even more (Kaput, 1998).
35
Research that provides insight into the way in which particular technologies can affect
student thinking may assist instructors in deciding the method of implementation of that
software. This can help them meet the needs of a diverse population of students with widely
varying social and cognitive needs. Providing greater research evidence is one of the factors that
can assist in this process, and help educators realize the potential of technology to improve
developmental mathematics instruction for all students (Epper & Baker, 2009). Developmental
educators would most likely benefit from evidence that comes from studies which bring
developmental mathematics students together with various types of technology, since a greater
base of evidence is needed to help them meet the challenges they face (Epper & Baker, 2009).
Such studies may help broaden the range of choice those educators have for their students. In
order to examine students? thinking in such a setting, an understanding of the role of
representation in mathematics education is necessary.
Representation in Mathematics Education
I will first consider how representation in mathematics education has been
conceptualized, beginning with a look at representational systems and the associated idea of
idiosyncratic representations (those which are unique to the learner) (Smith, 2003). Following
this is a look at other constructs related to representation, including visualization (the creation of
a mental image to guide the representation of ideas), and symbolization (the use of symbols to
organize the mind and reflect thoughts) (Moreno-Armella et al., 2008; Presmeg, 2006). A look at
something vital to the incorporation of representation in mathematics education, the use of
multiple representations for the same concept will lead to an examination of modeling and
functions. Modeling and functions are areas of mathematics in which the way representation is
used is particularly crucial. The systems of representation with which a student engages, their
36
visualizations, the connection of multiple representations, and the use of functions to model ideas
are all important considerations in the examination of technology use undertaken in the current
study.
Representation can be thought of as both the language of mathematics and the process of
illuminating ideas (Coulombe & Berenson, 2001). That is, it can be thought of as both a process
and a product. It refers to the act of representing an idea as well as to the form used for that act
(NCTM, 2000). Thought of this way, it permeates all of mathematics. Its effective use is a way
of both teaching and learning mathematics (Fennell & Rowan, 2001). Recent shifts in
educational practice include a ?heightened awareness of representation as a cognitive and social
process? as well as an increased understanding of the vital link it forms to knowledge (Monk,
2003, p. 250).
According to Cuoco and Curcio (2001), a representation is a map of correspondence
between a mathematical structure and a better understood structure. This map of correspondence
preserves the structure of what is being represented (Cuoco & Curcio, 2001). It ?re-presents? the
ideas so that a solution to a problem may be found (Smith, 2003, p. 263). Representations
facilitate reasoning and support different ways of thinking. They are the tools of proof and the
heart of communication (NCTM, 2000). Their effective use can provide classroom experiences
which can help students ?see the beauty and excitement in mathematics" (Cuoco & Curcio, 2001,
p. xiii).
Managing the long term process of conceptualization is more difficult in mathematics,
which encompasses a large variety of situations, procedures, and symbols (Vernaud, 1998). This
challenge makes understanding representation vital to mathematics education and representations
become much more than just ends in themselves. They become essential to understanding,
37
communication, justification, connections inside and outside of mathematics, and mathematical
applications (NCTM, 2000). Those that have researched the area of representation in
mathematics education have used a variety of different terms in their search to clarify its use. A
more detailed discussion of some of these ideas follows. For the readers? convenience, Table 7
lists some of the major contributors to the study of mathematical representation in chronological
order along with the major ideas contributed by the work listed. Some of those included were
contributors to influential collected works.
Table 7
Some of the major contributors to the study of representation in mathematics education
Author Year Major ideas
Kaput
1
1998 Functions are rich in representational possibilities
The phenomenon being represented should be at the
center of the study of functions. Theoretical framework
also presented, including ideas related to systems of
representation, notations, inscriptions, and language.
NCTM 2000 Presented and defined representation as a process
standard for teaching mathematics. Ideas include the
notion that representation is both a process and a product.
It refers to the act of representing an idea and to the form
used for that act.
1
Part of a two volume special edition of the Journal of Mathematical Behavior which focused on
representation and from which several other sources used herein were taken.
38
Author Year Major ideas
Yerushalmy and
Shternberg
2
2001 Described three representational phases involved in
learning algebra: graphic (a drawing of a situation), iconic
(sections of graphs seen holistically and used as icons),
and symbolic
Fennell &
Rowan
2001 Materials used to represent mathematical ideas may
replace student thinking rather then represent it.
Representations should not become ends in themselves.
Pape &
Tchoshanov
2001 Four implications for the use of representations as
cognitive tools rather then ends in themselves:
opportunity to practice, socialization, variety of
techniques, and use as tools for thinking, justifying and
explaining.
Goldin
3
2003 Described systems of representation and made
suggestions as to how researchers can examine students?
internal representations
2
From the 2001 yearbook of the National Council of Teachers of Mathematics, entitled The Roles of
Representation in School Mathematics edited by Albert A. Cuoco and Frances R Curcio. Several other sources used
herein were also taken from this source.
3
This, Monk (2003) and Smith 2003 are from A research companion to principles and standards for
school mathematics edited by Jeremy Kilpatrick, W. Gary Martin, and Deborah Schifter.
39
Author Year Major ideas
Monk 2003 Instead of being isolated items, graphs, charts, and other
representations can be tools for building understanding of
mathematics.
Smith 2003 Provided examples of and discussed students?
idiosyncratic representations. Discussed the importance of
helping students connect those representations to standard
ones. Conversations with students about how they
developed their representations can help teachers infer
students? internal reasoning.
Abramovich &
Norton
2006 ?Residual mental power? (p. 11) means that
representations developed during the use of technology
continue to be useful to the learner in the absence of
technology
Duval 2006 Connected semiotics to mathematics
Semiotic representations are tools for producing
knowledge
Presmeg 2006 Summary of research on visualization in mathematics
education. Ideas include the notion that students may not
have sufficient training with visual representations
40
Author Year Major ideas
Falcade,
Laborde, &
Mariotti
2007 Technology can allow access to more representations and
impact how mathematical objects are conceptualized
Semiotic mediation can allow new, internalized meanings
to be developed for mathematical tools
Moreno-Armella,
Hegedus, &
Kaput
2008 Mathematical notations, experiences, and the medium that
relates them co-evolve. Symbolic thinking has evolved
over time from static notations to dynamic technological
inscriptions.
Sedig, 2008 2008 Categorized representations into two broad categories,
textual (descriptive) and visual (depictive). Textual
representations are semantically dense, and conveyed
through rules. Visual representations are more analogical.
Representational systems within which learners operate. To better understand the role
of representation, it is helpful to consider the systems of representation within which the learner
operates. Those systems are generally described as either external or internal (Goldin, 2003). The
words external and internal refer to the relationship of that representation to the mind of the
student. If the representation exists within the mind of the student, then it is an internal
representation. If the representation is found in the environment outside of the student?s mind, in
a textbook, on a computer screen, or on a piece of paper for example, then it is considered to be
an external representation. External or internal representational systems are influenced by other
41
constructs such as the ideas of linguistics, affective notions, and habits of mind. An examination
of how some of these ideas relate to each other follows. A diagram summarizing some of the
ideas that will be discussed is provided in Figure 1.
Figure 1. The interplay between internal and external representations.
In 1998, two volumes of the Journal of Mathematical Behavior addressed representation.
The introduction to those two volumes, written by Goldin and Janvier (1998), described a system
of representation as referring to any of the following four categories of concepts. First, it can
refer to embodied representations of mathematical ideas which are external, physical situations
in the environment. Second, they said that it may refer to linguistic representations in which the
emphasis is on syntax and semantics. It may also refer to formal systems which use symbols,
axioms, definitions, constructs, etc. Finally, it may refer to internal, individual systems which
External, non-
standard
Idiosyncratic
representations
Internal
Mental
constructs
Visualizations
External,
standard
?Discipline
valued?
Symbols
42
describe thinking processes and are inferred from behavior or introspection (Goldin & Janvier,
1998).
Some researchers have also referred to the affective domain as a representational system
in and of itself. Goldin (2003) said:
The affective domain refers to feelings that pertain to mathematics, to the experiencing of
mathematics, or to oneself in relation to mathematics . . . affect serves a representational
function in the individual and . . . as a representational system, it enhances or impedes
mathematical understanding in certain ways. Local, changing states of feeling are not just
experienced but utilized by problem solvers and learners to store information, to
monitor, and to evoke heuristic processes (p. 280).
Frustration may signal the student to try a new strategy, signal that a problem is non-
routine, and provide impetus for a more effective approach. It may signal the possibility of joy at
meeting a challenge. In some students, frustration may invoke panic. Either way, it represents
something about mathematics to that student, either motivational or discouraging (Goldin, 2003).
Negative attitudes toward mathematics may be influenced by memories of past failures;
interactions with peers, teachers, and parents; and exposure to teaching methods, certain types of
mathematics, and certain learning environments (Sedig, 2008).
Representations occur externally in the physical environment or internally in the mind of
the person doing mathematics (NCTM, 2000). External representational systems include
conventional graphical and formal notational systems of mathematics, manipulatives, and
computer-based representations (Goldin, 2003). Internal representational systems are personal to
the learner and may consist of sensations, perceptions, imagined objects, or even emotional
feelings. They also include visual imagery, spatial, tactile and kinesthetic representations along
43
with students? personal conceptions and misconceptions. Each person forms their own internal
representational system (Goldin, 2003).
Mathematics is more than a collection of results and conjectures; it is also a collection of
habits of mind, such as those related to the learners? internal representational system (Cuoco &
Goldenberg, 1996). Students should develop the ability to visualize, describe, and analyze
situations mathematically (NCTM, 2000). One way that students understand a mathematical idea
is through reification, the process by which something abstract becomes real to the learner and
exists in his or her mind as a mental object (O'Callaghan, 1998). Students may, however, have
difficulty understanding standard representations which may be second nature to the teacher
(NCTM, 2000). Presmeg (2006) cited a study by Mourao (2002) which showed that students
may carry prototypical visual images, such as the image of a parabola with two real roots, which
are at odds with other, different but also valid visual images of the same concept. Such
insufficient internal representation can lead to manipulation of external representations without
attached meaning. Students with insufficient internal representations may cling to memorized
rules, rather than learning with comprehension (Saul, 2001). Representations students choose
themselves can help them understand and think through problems and bridge this gap of
understanding (Fennell & Rowan, 2001). They must be given the opportunity to practice
producing external representations and internalizing mathematical ideas (Pape & Tchoshanov,
2001). The internal and the external representational systems can then interact and represent each
other?s constructs in different situations (Goldin, 2003).
One of the challenges for researchers is that private representations and mental images
are hard to describe (Cuoco & Curcio, 2001). Goldin (2003) noted that such research relies on
observations of students? interactions with and production of external representations. He
44
proposed task-based interviews and suggested ten principles for planning, structuring, and
conducting them. Those principles include well-designed research questions and tasks, explicit
interview protocols with contingency plans, encouraging free problem-solving, and maximizing
interaction with the external environment through a variety of representational possibilities.
Smith (2003) analyzed the student's creative process through their language and beliefs in order
to see more than their external representations revealed. His conversations with them included a
discussion of the development of their representations. He concluded that research into
understanding how representations enable learning must proceed from inside the child (Smith,
2003). There is interplay between the external and internal representational systems; there are
many mental constructs which come into play in a discussion of the internal representational
system, and there are challenges involved in examining it.
Representations unique to the learner. As students work to connect the world of their
own internal representational system and the concepts within it to new mathematical ideas, they
may create their own unique, non-standard, idiosyncratic representations. These can help the
student cross the conceptual bridge to an understanding of standard representations. The way
students build representations is part of how they learn (Cuoco & Curcio, 2001). Mathematical
representations allow students to organize, understand, and communicate mathematical ideas
(NCTM, 2000). Monk (2003) has stated that ?students . . . have surprising representational
competence when their activity is within a coherent community with a sustained purpose? (p.
259). Helping students learn to choose the best representation wisely is vital. For example,
teachers must be familiar with the strengths and weaknesses of different types of representations
of a function, such as the quick reference aid a table provides and the global picture a graph
gives the viewer (NCTM, 2000).
45
One way of helping students develop their representational competence and assisting
them in learning to choose representations wisely is by incorporating the use of the students?
idiosyncratic representations (NCTM, 2000). NCTM encourages the use of representations
which are invented by and unique to the student (Smith, 2003). Students often generate
nonstandard representations for successfully accomplishing tasks (Izsak & Sherin, 2003). They
may also push the form of a representation beyond its original intentions (Monk, 2003).
Nonstandard representations often serve better than standard ones as tools for understanding and
communicating. They serve as bridges between a phenomenon and its standard mathematical
representation (Monk, 2003). Such personal forms of representation, which may be very
meaningful to the student, but have little resemblance to those commonly used, are an important
first step in students' developing the ability to use representations wisely. They may also
continue to be a tool with which students can reason and solve problems, even as their facility
with standard representations develops (NCTM, 2000).
One way to help students learn to use conventional representations is to engage the class
in whole group discourse about student generated representations (NCTM, 2000). Inventing
representations and then reflecting on them is a way to help promote fluency in the use of graphs
(Monk, 2003). Students? creation of idiosyncratic graphs, as opposed to standard ones, can be
followed by group consolidation onto a common graph. This can lead to recognition of the need
for standardization (Monk, 2003).
One goal in working with idiosyncratic representations is to help students make the
connection between their unique representations and discipline-valued representations. This is
necessary if they are to progress within the discipline of mathematics (Smith, 2003). Another is
to make inferences about student understanding. Inferences can be made about students' ability
46
to represent and understand mathematics by examining their language about their own
mathematical representations together with their attitudes towards their mathematics learning
(Smith, 2003). Some believe that the use of external representations of mathematical concepts to
develop student understanding is directly related to students? ability to visualize with those
representations (Pape & Tchoshanov, 2001). The ability to visualize would indicate that those
representations are part of their internal representational system (Goldin, 2003).
Idiosyncratic representations can be a valuable tool for the student. Operating with them
externally can lead to important classroom discourse and recognition of the need for
standardization. They can also help the teacher make inferences about student understanding.
Used externally, as they bridge to more conventional representations, they and the conventional
representations they lead to can both become tools with which the student can visualize
mathematics.
The role of visualization and imagery. Visualization is one of the constructs with which
researchers examine the field of representation, and it is an important consideration in examining
what internal representations students may possess. Visualizations and imagery may be
categorized in different ways as seen in the research discussed below.
Some researchers categorize representations broadly as either textual (descriptive) or
visual (depictive). Textual refers to those representations which are like language, semantically
dense, and conveyed through rules. Visual representations, as opposed to textual, are more
analogical in nature (Sedig, 2008). Smith (2003) cited Goldin and Kaput (1996) as stating that
imagistic representations include internal imagery which may sometimes demonstrate meaning
through visualization, analogy, or metaphor.
47
Presmeg (2006) used the term visualization to describe the creation of a visual image
(mental construct) which guides the creation of mathematical inscriptions (representations). She
referred to those who prefer to use visual methods as visualizers. Monk (2003) described seeing
as a constructed activity, closely linked to thoughts and actions. How people see things is linked
to how they think about them. Changes in the way students see things can be fostered. For
example, many visual displays can be used, rather than a single visual representation, such as a
traditional graph (Monk, 2003).
Presmeg (2006) listed five types of imagery used by high school learners: concrete
imagery, kinesthetic imagery, dynamic imagery, memory images, and pattern imagery. Concrete
imagery is a picture in the mind. Kinesthetic imagery refers to physical movement. Dynamic
imagery occurs when the image itself is moved or transformed. Memory images are those which
allow learners to recall formulas. Pattern imagery is "pure relationships stripped of concrete
details" (Presmeg, 2006, p. 210). Different categories of imagery may overlap. Visualization is a
powerful tool in algebra as well as geometry and trigonometry, however, students may not have
sufficient training with visual representations. For example, dynamic imagery was shown in one
study to be used effectively, but rarely, by high school students (Presmeg, 2006). Textual, visual,
concrete, kinesthetic, dynamic, memory, and pattern images have all been proposed as possible
ways in which students visualize mathematics. They can be considered as ideas with which to
examine a student?s internal representational system.
Symbolization as a construct related to representation. While visualization is directly
related to a student?s internal representational system, external representations are more closely
related to the idea of symbolization. External representations may be idiosyncratic, but more
often they employ standard mathematical symbols. The idea of symbolization is another lens
48
through which researchers have examined representations. An examination of research shows
that symbols have been discussed as representing concepts, organizing the human mind,
fostering the sharing of ideas, providing a link to abstract ideas, and becoming a new internalized
tool with which the student can build new knowledge.
Idiosyncratic, internal, and external representations each may provide different instances
of one idea. A unifying expression for a set of multiple instances requires some sort of symbolic
structure (Moreno-Armella et al., 2008). Symbols represent items from a reference field. For
example, one reference field for the set of nouns is the set of material objects. Reference fields
grow and transform with the shared use of symbols (Moreno-Armella et al., 2008). Symbolic
structures re-design the architecture of the human mind and provide a meta-cognitive mirror in
which our thought is reflected (Moreno-Armella et al., 2008).
Educators sometimes refer to such a process as symbolization. Each mathematical
representation stands for some mathematical concept. In the process of symbolization, symbols
and referents are sometimes experienced as separate items and sometimes experienced as the
same thing (Kaput, Blanton, & Moreno, 2008). Symbolizations can be privately constructed and
used by one individual, as with students? idiosyncratic representations. They can also be shared
by a community, as with commonly used external representational systems, which may be the
product of a long process of refinement (Kaput et al., 2008).
Symbols may sometimes be treated as objects in their own right without regard to the
referent for which they stand. Whitehead (1929), as quoted by Kaput, Blanton, and Moreno
(2008) stated, "Civilization advances by extending the number of important operations we can
perform without thinking about them" (p. 22). Students may operate on symbols by following
rules and algorithms. Using this method of operation, students may, indeed, be acting upon the
49
marks on the paper without understanding. They may act based on the position of the symbols,
not based on their meaning. On the other hand, writing which is not based on rules for
manipulating symbols may be used to develop an idea or build an argument. In such instances,
students may draw diagrams, for example. If so, what they write is not determined by "strict
syntactically defined rules" (Kaput et al., 2008, p. 24).
Symbolization cannot be separated from conceptualization. Rather than promoting the
lack of thought, it has the potential to provide a "lift-off" from concrete thinking (Kaput et al.,
2008, p. 23). Algebraic symbolization historically gave rise to the understanding of new ideas
such as negative and complex numbers. Through symbolization, "new mathematical worlds
become possible" to students - one possible reason why algebra serves as a gateway to further
mathematical development (Kaput et al., 2008, p. 23). The manner in which mathematical
discourse and mathematical objects interact is a creative and continuous process, involving
symbolization, which has occurred throughout history. It continues to occur in classrooms as
well as in the minds of individuals (Kaput et al., 2008).
Symbols also serve as signs. Semiotics is the study of signs and their meanings.
Cunningham (1992) described semiotics as "a way of thinking about the mind, and how we come
to know and communicate knowledge" (p. 166). He also noted that it has an "ecumenical nature"
in that it "draws from" and "informs" many other disciplines (Cunningham, 1992, pp. 166-167).
He argued that "knowledge does not exist separate from the knower" and that "knowing can't be
anything but personal knowledge" (Cunningham, 1992). This is not to say that reality does not
exist, but that our understanding of it is constructed, and there are limitations on those
constructions determined in part by the existing structure of our thoughts. Though different
50
individuals? constructions of understanding may differ, similarities across individual
constructions may reveal something about reality.
Representations can be considered signs, and their associations are produced according to
a set of rules, allowing the description of a system. In this system, the semiotic representations
become tools for producing new knowledge, not just for communicating a certain internal
representation (Duval, 2006). Semiotic mediation refers to the use of a semiotic system or tool in
social interaction so that new signs are generated to foster internalization of meanings. Under
proper guidance, new meanings related to the use of a tool can be formed and developed
(Falcade et al., 2007).
Internalization occurs through systems of signs and semiotic processes. A tool used under
expert guidance to accomplish a task functions as a semiotic mediator as new signs are derived
from the actions performed with the tool. This fosters an internalization process, producing a
new internal tool. The internal tool may resemble in some respects the actual external tool and
new meanings may be generated related to the use of the tool (Falcade et al., 2007). This
research gives us greater insight into the role played by those symbols normally thought of as
forming the body of tools for representing mathematics. It is possible to see that they can take on
added meaning, producing new knowledge as well as communicating what is already known.
Connecting multiple representations. It was noted in the discussion of symbolization
that seeking a unifying way to express multiple instances of the same idea may lead to the
development of symbols. In a similar manner, those who encounter multiple symbols or
representations for the same idea may find their understanding of that idea deepening. Students
who learn to move flexibly among and choose from a variety of representations to solve
problems will also be deepening their understanding of mathematics.
51
One way students demonstrate conceptual understanding is by their ability to move from
one representation to another and coherently use different representations (Even, 1998; Hitt,
1998; O'Callaghan, 1998). As the number of representational tools a student uses expands, he or
she will need to move flexibly among different representations in order to view mathematical
ideas from the different perspectives those tools offer. This ability will enhance their
mathematical power (NCTM, 2000). Those who are able to switch representations can use the
representation most beneficial to their analysis (Even, 1998). Mathematically proficient students
productively choose from a selection of possible representations in order to solve problems
(NCTM, 2000). In addition, the combined use of different types of representations has the
potential to cancel disadvantages each type may have. They may provide greater information
when used together (Friedlander & Tabach, 2001). Monk (2003) asserted:
The goal is not to select one or two representational forms for students to learn and use in
all situations but, rather, to teach students to adapt representations to a particular context
and purpose and even to use several representations at the same time (p. 260).
The use of multiple diverse representations has been shown to be important to the
understanding of students at higher levels of mathematics (Santos-Trigo, 2002). Instead of being
isolated items, graphs, charts, and other representations can be tools for building understanding
of mathematics (Monk, 2003).
Students may resist transitioning between different representations (Friedlander &
Tabach, 2001). To help encourage them, tasks can be designed which promote the use of
multiple representations. Such tasks may promote frequent transitions between representations,
and the use of different representations becomes a "natural need" rather than an "arbitrary
requirement" (Friedlander & Tabach, 2001, p. 176). An environment where teachers and students
52
are presenting in different representations encourages flexibility in the choice of representations.
Students may be started with problems requiring the use of a specific representation and then
later be asked more open-ended questions (Friedlander & Tabach, 2001). As teachers incorporate
multiple representations into their classrooms, student thinking will emerge and their erroneous
ideas can be addressed (Hitt, 1998). Teachers help students progress from seeing representations
as ends in themselves to the act of representing by engaging them in ongoing discussions of the
reasons for choosing one kind of representation over another (Monk, 2003).
There is potential vulnerability in making conceptual and representational connections
(Hitt, 1998). Two representations may be mathematically equivalent but cognitively non-
equivalent in that they are processed differently by different learners. Presenting ideas in ways
that convey the most meaning to the most students will bring maximum benefit. This may mean
using such visual cues as graphic indicators of number size, the use of proximity and color to
distinguish related items, and the placement of related representations so they can all be seen at
once (Gadanidis et al., 2004).
As students grow and learn about mathematics their use of representations grows from
directly perceived objects and actions, to indirectly perceived items such as rational numbers,
and eventually to abstract ideas such as functions (NCTM, 2000). Abstraction in mathematics is
the stripping away of features not necessary for analysis. By facilitating abstraction,
representational ability aids students in identifying common underlying mathematical structures
which appear in different settings. It also allows them to examine essential features of problems
and their mathematical relationships. Rich representations allow students to examine many
aspects of this process (NCTM, 2000). It has been said that the student use of multiple
representations in working with the same mathematical object is what constitutes ideal
53
mathematics learning (Hsieh & Lin, 2008). Choosing, adapting, transitioning, communicating,
abstracting, and justifying are all aspects of the use of representations for the study of
mathematics. Each of these habits has the potential to emerge in the presence of multiple
representations.
Uses of representations: Models and functions. Two fruitful and closely related areas
for the study of how representations affect mathematics are modeling and functions. Models and
functions both can be used to illustrate real-life phenomenon mathematically and both regularly
involve the use of multiple representations. A look at models as a form of representations will be
followed by a look at representational ideas associated with functions.
Models as representations. One form of study in the field of mathematics which employs
representation as a problem solving tool is modeling. Models link different ideas together;
particularly they link concepts outside of mathematics to mathematics. The process of modeling
helps students to notice underlying mathematical structures in the world around them, builds the
idea of isomorphism (a one to one relationship between two sets of data preserving operations
within the two sets), broadens their understanding of what it means to represent something, and
deepens their understanding of mathematics (Abrams, 2001).
A mathematical model is a form of representation which illustrates mathematical features
of a complex phenomenon and is used to clarify situations and solve problems (NCTM, 2000).
Models have the potential to provide an important service, since cognitive knowledge is closely
linked with the knowledge people have of a situation being represented (Monk, 2003).
Traditional word problems have usually involved the use of a specific formula or algorithm, and
an easily detectable list of data to be used. This is different from authentic problem solving using
modeling tools (Yerushalmy & Shternberg, 2001).
54
Abrams (2001) defined modeling as the process of studying questions outside of
mathematics with mathematics. When used for modeling real life situations, mathematics serves
as an "intellectual lens" for examining questions (Abrams, 2001, p. 269). In this sense it is not
self-contained, but used as a tool in other disciplines, as well as for abstract discoveries. When
mathematical modeling is ignored, some skills, including choosing appropriate representations
for a situation, and recognizing common structures are neglected (Abrams, 2001). As students?
understanding and use of representations develops and becomes more sophisticated they can
learn to use variables, tables, equations, and graphs to model and analyze real life phenomenon
(NCTM, 2000). Modeling can be supportive of emerging representations of functions
(Yerushalmy & Shternberg, 2001).
Modeling involves examining two ideas with matching structures, which builds the
concept of isomorphism (Abrams, 2001). It maps the real-life situation to its mathematical
model. The modeling cycle consists of posing a question, selecting the representation(s),
creating a model, manipulating the model, determining mathematical products, translating
(interpreting mathematical results according to the setting), deriving new knowledge, and
analyzing the results. Analysis leads back to the question until a sufficient model is created
(Abrams, 2001). As part of the selection and creation of the representation, a real situation may
first be idealized into a pseudo-concrete model before it is further abstracted into a mathematical
model (Presmeg, 2006) .
If students are allowed to interpret familiar events mathematically, then they can
understand representation more deeply (Coulombe & Berenson, 2001). They can also more
easily access problems which can be represented in ways that are meaningful to them (Fennell &
Rowan, 2001). Some students may, in the process of representing a real life situation, strip the
55
context away while others may use the context. Stripping the context away seems in some cases
to have helped the student avoid confusion (Smith, 2003). A deeper interpretation of
mathematics based on familiar events and problem solving can broaden students? understanding
of conventional representations beyond mere manipulation (Coulombe & Berenson, 2001). The
process of modeling helps develop the mathematical lens through which a student views the
world. That modeling process may take several steps through different levels of abstraction.
Representations, both idiosyncratic and standard, can be part of that process along the way.
Through modeling, students can improve their facility with representational tools.
Function as a context for studying representation. Functions are often used as a
mathematical context for studies of representation because they are rich in representational
possibilities. For example, the encouragement to regularly immerse students in mathematical
experiences which involve an interplay of symbolic, numerical, and graphical forms of
representation is commonly known as the rule of three (Reinford, 1998). The rule of three was
meant to encourage students to take facts they see in graphs and verify them numerically and
algebraically (Bridger & Bridger, 2001). Functions can be represented all three ways. Function is
also considered by many to be the most important concept in all of mathematics and fundamental
to the learning of mathematics (Hitt, 1998; O'Callaghan, 1998). The concept of function has been
important in the study of students? obstacles with regard to representations (Goldin, 2003). The
ability to interpret and translate representations can help students construct their mental images
of patterns and functions and thus extend their algebraic thinking (Coulombe & Berenson, 2001).
Even (1998) referred to two ways participants in one study dealt with functions:
pointwise or globally. To deal with functions in a pointwise way is to plot, read or deal with
discrete points. To deal with the function in a global way is to look at its overall behavior, such
56
as when students sketch the graph of a function and look at its maximums and minimums and
other characteristics. This study suggested that those who can easily use a global analysis of
changes in the graphic representation of a function have a better understanding of the
relationships between graphic and symbolic representations than those that check local
characteristics (Even, 1998).
When the student is able to understand the different representations of functions, those
representations can serve as windows into functional relationships in particular situations (Lloyd
& Wilson, 1998). Representations of functions then become valuable tools for modeling real life
situations. Students having a good concept of function will be aided in solving problems
(Yerushalmy & Shternberg, 2001).
Numerical representations are some of the first representations that students encounter,
and provide some of their first experiences forming internal representations (Pape &
Tchoshanov, 2001). At first, children may not understand that when they are counting, the last
word they say represents the number of things they have counted all together. Eventually the
name of the number comes to represent a set of that many objects (Pape & Tchoshanov, 2001).
Later uses of numerical representations are familiar and convenient, but lack generality. This
limits some of their problem solving potential (Friedlander & Tabach, 2001). It is not directly
evident that when teachers use manipulatives to represent number concepts, such as base ten
blocks used for learning regrouping, that students see the connection between the manipulatives
and the mathematical activities they are intended to represent. Some believe that the use of such
external representations of numbers to develop a student?s understanding of mathematics is
directly related to the student?s ability to visualize with those representations (Pape &
Tchoshanov, 2001). Algebraic representations have many advantages, such as conciseness,
57
generality, and effective modeling. Sometimes, they provide the only avenue of proof. The
problem comes when their exclusive use interferes with the conceptual understanding of what
they represent (Friedlander & Tabach, 2001).
Verbal representations assist in understanding context, communicating results, solving
problems, and working with patterns (Friedlander & Tabach, 2001). They emphasize connections
to other domains of study. Verbal representations can, however, suffer from ambiguity and thus
they can possibly become obstacles to communication (Friedlander & Tabach, 2001). Oral and
written language as tools of communication can be considered forms of mathematical
representation (Coulombe & Berenson, 2001).
A graph can be thought of as a lens through which to explore the phenomenon graphed.
Learning to read graphs is referred to by some researchers as ?disciplined perception? (Monk,
2003). Graphs connect formal static definitions of function with the metaphor of motion (Falcade
et al., 2007). One of the complexities in graphing as a representation is that a graph has many
potential meanings. Those who are fluent graph readers can forget the difficulties others have
(Monk, 2003). Inappropriate responses to visual attributes of a graph are the most frequently
cited student errors. Some mathematics educators have begun to focus on the process by which a
learner constructs meaning from a graphical representation in addition to focusing on the
information itself (Monk, 2003).
A graph does not reach its full potential until it is used to make meaning. Monk (2003)
suggested a variety of meaning making processes to aid students in making meaning from
graphs, which can be transferred to the use of technological representations. Students can explore
aspects of the situation graphed that were not otherwise apparent. The process of representing a
context can lead to questions about it. Graphing and analyzing a well understood concept can
58
help them understand graphing better. Important features of a graph help them construct new
concepts. Understanding of the graph and the context can build at the same time. Finally, a group
can build shared understanding through the common use of a graph as a window into a
phenomenon (Monk, 2003).
Another approach to the study of functions in addition to the traditional Cartesian plane is
to view them via mapping diagrams (Bridger & Bridger, 2001). Bridger and Bridger?s (2001)
use of function mapping diagrams is taken from the science of map projection, which renders an
image of a globe on a flat surface. A function mapping diagram renders the x and y axes as
parallel lines, with line segments connecting a point on the x-axis to its image on the y-axis.
Different representations of functions have different advantages and disadvantages (Bridger &
Bridger, 2001). Cartesian graphs are more useful for examining extrema, convexity, and
asymptotes. Advocates of the function mapping approach feel that traditional graphs may inhibit
the development of a mapping concept of functions (Bridger & Bridger, 2001). They point out
that, in addition to promoting the concept of function as a mapping, they also show whether the
function is an expansion or a contraction, and where and how it is one-to-one or many-to-one.
They are also excellent for visualizing compositions and inverses of functions (Bridger &
Bridger, 2001). Both models and functions, which are closely related, overlapping mathematical
ideas, give us ample opportunity to consider the effect of representation on student
understanding. In examining that understanding it will be helpful to consider an interpretive
framework.
Building validity, usefulness, and endurance. One of the ideas discussed in regard to
representation was the concept of a student?s internal representation, which incorporates the idea
of conceptualizing and symbolizing. In considering the relationship between a student?s internal
59
representational system and their success in learning mathematics, I posit that it is useful to
organize key ideas into three constructs: validity, usefulness, and endurance. These constructs,
summarized in Table 8, form an interpretive framework with which the effect of technology on
student learning can be examined.
Table 8
Interpretive framework for ideas related to representation
Representations are
Valid if they Useful if they Enduring if they
Accurately reflect the
mathematics they seek to
represent and are flexible enough
to allow additional mathematical
ideas to be built upon them. Are
accompanied by sound
mathematical habits of mind.
Are accessible for
reasoning and sense-
making, communication of
mathematical ideas, and
building new
understanding.
Remain with the student in
various situations apart from
the environment in which they
were initially developed. Are
carried forward, built upon,
and refined over a period of
time.
Valid internal representations. Validity in research allows that research to be correctly
interpreted (Gay, 1996). In mathematics, a representation may be considered valid if it accurately
represents the mathematics it seeks to represent and is flexible enough to allow additional
mathematical ideas to be built upon it. Not all internal representations are mathematically valid.
Research has shown that students may hold prototypical images of mathematical concepts, but
that these images may limit their thinking and force them to rely on memorization, providing
very little true comprehension (Saul, 2001). One such prototypical image is the parabola with
two real roots, which, when fixed in the student?s mind, impedes the idea that a quadratic
60
equation may have just one real root or none at all (Presmeg, 2006). This concern is related to the
thoughts expressed by Rogers (1999) that simplified diagrams make take away from the deeper
learning that other representations may provide. Prototypical images which provide no true
comprehension are not accurately representing the mathematics and simplified methods which
short-cut deeper understanding are too inflexible to build upon.
Part of the validity of mathematical representations is situated in the mathematical habits
of mind which accompany them and which assist the student in translating them into other valid
representations (Cuoco & Goldenberg, 1996; Even, 1998). The parabola with two real roots may
be invalid for a student who has no accompanying habits of mind allowing the switch to one or
zero real roots. It may, however, be valid for another student who does possess those habits of
mind. In addition to this type of flexibility, mathematically valid representations do not merely
represent each other, they also represent contextual situations or reified concepts (Kaput, 1998;
O?Callaghan, 1998). Such conceptualizations indicate that the mathematics is real to the learner
and accompanied by understanding (O'Callaghan, 1998).
Useful internal representations. Representations are useful when they can be selected
and applied to maximize problem solving, build new knowledge, and communicate mathematical
ideas to others. Students demonstrate mathematical proficiency and power when they can use
representations in these ways. The representations they hold are not useful to them if they cannot
communicate and solve problems with them (NCTM, 2000). Note, for example, that there is an
important difference in the learning process when a teacher uses representations as part of a
dynamic, active process, which facilitates sense-making as opposed to making explicit the
representations with which they wish their students to solve problems. The former involves
students in the act of re-presenting to themselves prior mathematical activities in ways crucial to
61
the knowledge they are currently constructing (Cifarelli, 1998). This sense-making process
requires useful representations as tools for students. They are available as what Cifarelli (1998)
called ?interpretive tools of understanding? (p. 241). They are general enough in the mind of the
student so that their problem solving potential is not limited (Friedlander & Tabach, 2001).
Using a problem based approach in teaching can give students the opportunity to construct and
interpret graphs, generate data, find patterns, and interpret mathematical ideas in other ways
(Coulombe & Berenson, 2001). This can allow the teacher to discover something about the
usefulness of the student?s internal representations. Koedinger and Nathan (2004) referred to the
process of using familiar representations to build new ones as ?grounding? (p. 158). Useful
representations allow students to ground their new understanding in that which they already
know. They will also be available for students so that they can organize, record, and
communicate mathematical ideas (Fennell & Rowan, 2001). Useful representations may be non-
standard, but non-standard representations often serve well as tools for understanding or
communicating (Monk, 2003). Useful internal representations provide students with
mathematical power, which includes the ability to reason, communicate, discover, conjecture and
connect mathematics within itself and outside itself (NCTM, 1991).
Enduring internal representations. Internal representations may be considered enduring
if they stay with the student in various situations apart from the environment in which they were
developed and are carried forward to later work in which they are deepened and built upon as the
student progresses mathematically. Abramovich and Norton (2006) referred to representations
which may develop during the use of technology, but continue to be useful to the learner in the
absence of technology as providing ?residual mental power? (p. 11). In their study, the use of a
locus approach for solving problems with parameters, studied with the use of technological
62
representations, resulted in the locus becoming a familiar mathematical thinking device
(Abramovich & Norton, 2006). The variety of symbols used in mathematics makes long term
conceptualization challenging for some students (Vernaud, 1998). A vivid and meaningful
representation can be held onto, built upon, and refined over a period of time within an
individual, thus aiding long-term conceptualization (Kaput, Blanton, and Moreno, 2008).
Enduring representations become part of what Rogers referred to as ?stored knowledge? which is
part of the critical process of knowledge integration and reasoning (Rogers, 1999).
Cifarelli (1998) noted that representation has a constructive function which involves the
development of mental objects which students can reflect on and transform. Enduring
representations will be stored and available to the student for future work and remain part of the
set of mental objects upon which they reflect in order to build new knowledge.
Building valid, useful, and enduring internal representations of mathematics within
students will provide them with clarity of understanding, power in problem solving and
communication, and a continually developing storehouse of mathematical knowledge.
Considering these representational ideas will provide a framework with which to examine the
interplay between technology and student learning. Following is a closer examination of the
interplay between technology and representation.
The Connections Between Technology and Representation
The connections between technology and the use of mathematical representations are
many. Technology transforms the possibilities present in mathematical representations (Moreno-
Armella et al., 2008). It can allow students to gain a better understanding of the use of
representations (Maximo & Ceballos, 2004). Kaput (1998) believed that technology had the
63
potential to build grounded understanding of mathematical ideas so that different representations
are doing more than just representing each other.
Among the ways technology and the use of mathematical representations influence each
other is the ability of technology to link multiple representations of the same phenomenon
(Hennessy et al., 2001). Technology also allows a connection between real-life phenomenon and
the representations which depict them (Stylianou et al., 2005). In addition, technology can allow
students to access a variety of different types of representations and develop a deeper
understanding of representations (Alagic & Palenz, 2006). Technology allows students to use
representations to explore problems that were previously inaccessible, such as the modeling of
complex real-life phenomenon and the examination of multiple changes in mathematical
parameters (NCTM, 2000). Technology also allows a direct link between a real life phenomenon
and its associated representations through the use of calculator based laboratories. In this way,
using computer technology provides for bidirectional interactions (the phenomenon affects the
representations and this affects the phenomenon), and this exchange can become rapid, allowing
student hypotheses to be quickly tested (Kaput, 1998).
Technology increases the opportunity to analyze multiple, connected representations
(Yerushalmy & Shternberg, 2001). Heid and Blume (2008) noted technology's increasing
sophistication and and "multirepresentational capability" (p. 58). Technology can connect
mathematics with visual representations in such a way as to foster mathematical thinking and
conceptual understanding (Lopez Jr, 2001). Through the promotion of multiple representations,
technology imparts the advantage of flexibility to visual reasoning (Friedlander & Tabach, 2001;
Presmeg, 2006). Graphing calculators can mediate students? problem solving by providing
64
seamless switching between symbolic, graphical, and numerical representations (Hennessyet al.,
2001).
Microsoft Excel's "ability to integrate multiple representations" helps students think of
things in different ways. Students must also think carefully about the role of variables which is
vital to their understanding of algebra (Donovan II, 2006). Interactive diagrams can allow
multiple representations to be combined and changed in relationship to each other. Rogers (1999)
called this process "dynalinking" (p. 423). When a computer simulation of a real-life setting was
linked to an interfaced diagram showing aspects of that setting, students were able to learn more
from working with the interfaced technological diagram than they had learned with static
textbook diagrams. Their ability to reason with the abstract diagrams had improved (Rogers
1999). In the same way, through the use of technology graphs have become manipulable,
whereas they were previously seen as static, fixed entities (Kaput, 1998).
Technology allows students access to representational ideas that otherwise might be
difficult to share or visualize, such as a depiction of the movement of two variables at the same
time (Falcade et al., 2007). Technology may allow students to make predictions about what a
representation is telling them and then test their conjecture, allowing them to build increasing
understanding of that representation (Hegedus & Kaput, 2004). Students may use technology to
create different representations, leading them to make their own mathematical conjectures
(Santos-Trigo, 2002).
Technology can also assist students in developing greater conceptual understanding of
representational ideas such as algebraic symbolization (Abramovich & Ehrlich, 2007). For
example, they are often are able to solve equations with no conceptual understanding of what
they are doing, but that lack of understanding is a much greater handicap when solving
65
inequalities. Technology can allow them to examine a series of comparisons which can give
them insight into the effects of different choices of algebraic manipulation on the solution set of
inequalities. This can give them a conceptual underpinning to what otherwise might be an
ungrounded set of rules and procedures (Abramovich & Ehrlich, 2007).
In considering these connections and the use of technology in the classroom to enhance
the use of mathematical representations, teachers may wish to have certain things in mind. For
example, having students produce their own representations of what is occurring within the
technological representation may be important in making explicit the implicit knowledge that
teachers may take for granted (Hennessy et al., 2001). Knowledge called for with the use of
technological representations includes an understanding of the method of input and mathematical
interpretation of output associated with the formats the technology uses. This may include
notation which varies from the notation seen in textbooks. The use of such technology in the
classroom requires the teacher to carefully consider the role of representation. (NCTM, 2000).
Following is an examination of several studies which examine the use of both technology and
mathematical representations. Table 9 provides a summary of the some of the major studies
discussed.
66
Table 9
A selection of studies in technology and representation
Aspect of
technology
Researchers What was done What they found
Using relations
to look at algebra
from a
geometric
perspsective
Abramovich
& Norton
(2006)
Abramovich
& Erlich
(2007)
Pre-service teachers were
engaged with graphing
software which allowed
relations from any two-
variable equation to
graphed
2006: Participants
better understood
connections between
geometry and algebra
2007: Visualization
was fostered which
provided conceptual
insight.
The use of
spreadsheets as
cognitive tools
Alagic &
Palenz (2006)
Used Microsoft Excel for
professional development
with middle school
mathematics teachers.
Teachers explored a real-
world problem using
multiple representations.
Teacher were able to
explore many ideas in
a small amount of time
67
Aspect of
technology
Researchers What was done What they found
The use of
spreadsheets as
cognitive tools
Hsieh & Lin
(2008)
Teaching experiment
involving 8 sessions with
three fifth grade students
who needed mathematical
remediation
Linked representations
produced greater
progress in
understanding and
knowledge was
transferred to internal
representations
The use of object
oriented
programming to
build self-
efficacy
Connell
(1998)
Stevens, To,
Harris, &
Dwyer (2008)
52 caucasian lower-middle
class elementary school
students in student centered
classrooms were taught
object oriented
programming language for
mathematics ? one class for
presentation by the teacher
and for exploration
Gifted children worked
with LOGO computer
software
Students benefitted
from using the
computer as a
reflective tool which
reacts to student input
in a way that
encourages accuracy
Gifted children
working with LOGO
increased in creativity
and verbal
mathematical ability
68
Aspect of
technology
Researchers What was done What they found
Exploring
connections with
graphing
calculators and
dynamic
software
Hennessy,
Fung, and
Scanlon
(2001)
Examined adults working
with each other on activities
involving graphing
calculators
Advantages: speed,
visualization,
movement among
representations, need
to clarify ideas, pairs
problem solving
mediated
Disadvantages:
assessing mechanical
vs. conceptual
knowledge
Paper and pencil use
encouraged
69
Aspect of
technology
Researchers What was done What they found
Exploring
connections with
graphing
calculators and
dynamic
software
Santos-Trigo
(2002)
Yerushalmy
& Sternberg
(20010
25 first year university
students in a calculus class
for which calculators and
dynamic computer software
were available were
interviewed and written
reports were gathered
related to three tasks
Created Function Sketcher
software for use with
seventh grade students to
take them through
algebraic, graphic, iconic,
and symbolic phases to
develop the concept of a
function
A visual approach
helped the students
understand the nature
of the roots of
equations. They were
able to make
connections between
representations.
Students could see
how to translate a real-
life event into a
graphical
representation and a
graphical
representation into a
symbolic one
70
Aspect of
technology
Researchers What was done What they found
Linked whole
group study
Hegedus &
Kaput, 2004
MathWorlds software used
in teaching experiments in
middle and high school
classrooms in which each
students or groups work
could be uploaded and
chosen for display to the
class
Displaying student
work helped focus
attention on the
underlying
mathematical
structure. Students felt
personally connected
to the work presented
The use of
dynamic
geometry
environments to
deepen thinking
Cuoco &
Goldenberg,
1996
Falcade,
Laboorde,
and Mariotti,
2007
Provided samples of tasks
from studies done in
computational and dynamic
geometry environments
Used the trace tool in a
dynamic geometry
environment to introduce
the concept of functions by
looking at the covariation
of dragged objects
Mathematical objects
seemed to become real
and the object of
experimentation
The teacher?s role is
important in helping
students make
meaning. Students?
work can form the
basis for discussion
71
Recent Studies in Technology and Representation
Recent studies which focus on both technology and representation are numerous and
varied in their approach. Following is a brief look at several studies conducted over the past ten
years. The topics examined by these studies include the development of software to re-examine
algebra through the graphing of relations, using spreadsheets as cognitive tools, the use of object
oriented programming to develop logical thinking, looking at dynamic connections between
representational forms with calculators and computers, the use of data-collecting laboratory
tools, the use of technology to link a classroom of students in whole group study of their work,
and the use of dynamic geometry software to deepen mathematical thinking.
Using relations to look at algebra from a geometric perspective. Abramovich?s two
studies, one with Norton (2006) and one with Erlich (2007) demonstrated the potential of
technology to assist students in gaining a deeper understanding of inequalities. His work
provided insight into errors through the use of visualization made possible through the use of
graphing technology. Graphing software was developed which was able to graph a relation from
any two-variable equation, as opposed to the standard calculator technology which requires the
equation to be solved for the dependent variable. Both studies were done with pre-service
teachers. The 2006 study allowed the participants to better understand the use of geometric ideas
to understand algebraic relationships. The 2007 study showed that the graphing technology
fostered visualization which gave the participants conceptual insight. They were able to see how
algebraic manipulation affected the solution set of an inequality.
The use of spreadsheets as cognitive tools. Alagic and Palenz (2006) used Microsoft
Excel (Excel) in a professional development setting involving middle school mathematics
teachers. Their study focused on the development of conceptual understanding which can come
72
from the use of multiple representations, using spreadsheets as cognitive tools. Their professional
development model combined the immersion of teachers in the exploration of real-world
problems and the connection of those activities to their classroom work. The teachers examined a
problem involving exponential data from a story problem (The King?s Chessboard). They looked
at the data in tabular form, graphed it, and zoomed in and out to see how the appearance of the
graph changed when the window was changed. The ability of teachers to distinguish between
exponential and linear growth was increased as a result. The teachers then created their own
stories and activities for their students. The technology allowed the teachers to explore a variety
of instances of a mathematical idea in a smaller amount of time. Hsieh and Lin (2008) also used
Excel for a study involving multiple representations. They conducted eight sessions of a teaching
experiment with three fifth grade students who needed remediation in mathematics. Their
subjects had no difficulty with reading, but did have difficulty decoding textual material related
to mathematics. The Excel based lessons the researchers provided included textual, numerical,
and graphical representations of word problems and provided students with instant feedback with
which they could observe changes in representations resulting from their choices. The
researchers found that such representations, when linked, resulted in greater progress in
understanding. The students? knowledge was transferred to internal representations which
allowed them to solve new problems (cf. p. 230 #6).
The use of object oriented programming to build self-efficacy. Connell (1998)
discussed the use of an object oriented computer authoring language to create personally
meaningful representations by means of computer based tools in a constructivist environment.
Connell (1998) noted that the object oriented qualities of the software used in his study allowed
the students to use powerful graphic tools such as drawing implements in a setting requiring
73
relatively simple syntaxes to create their programs. 52 predominantly Caucasian lower-middle
class elementary school students in two rural elementary school classrooms were the subjects of
this study. Their teachers had been observed using constructivist methods (student centered,
facilitative, encouraging problem solving) prior to the study?s beginning and had had a year?s
experience with the materials used in the study. Though the technology was used in both
classrooms, it was only used for student exploration in one of the classrooms. In the other it was
used as a presentation tool. Students working with the technology in this study could create their
own personal representations and tools, which required them to write programming scripts which
would perform as desired. Results showed that the students benefited from computer use which
goes beyond ?delivery of static information? and acts as a reflective tool which reacts to student
input in a way that encourages mathematical accuracy.
One report looked at the effect of study with LOGO computer software on seventh grade
teachers? self-efficacy and self-determination (Stevens, To, Harris, & Dwyer, 2008). LOGO is
computer software providing a graphics programming language which allows the student to write
directions for programming activities and receive immediate visual feedback. It was chosen for
its ability to take basic concepts to more complex levels. The problem solving process involved
in using LOGO can offer a "window into the student's mind" (Stevens et al., 2008, p. 199). The
student must apply logical reasoning to their programming. Researchers have recommended that
the use of such software be facilitated by teachers who can help students make connections with
mathematical ideas and recognize their own thinking processes. It has also been suggested that
such software can help students learn to work through challenges with more confidence. Failure
becomes an "opportunity to plan a new course of action" (Stevens et al., 2008, p. 200). Gifted
children working with LOGO were shown to increase in creativity, and in verbal domains of
74
mathematics. The teachers in this project appeared to be encouraged to implement technology
into their classrooms and work through obstacles that might arise as they attempted to do so.
In a related effort Kynigos, Psycharis, and Moustaki (2010) conducted an experiment
with eight 17 year old students studying mechanical engineering in a secondary technical and
vocational school. The students were instructed in the use of MoPiX, a computer software
environment in which formal algebraic structures were used to manipulate animated models of
real-life situations. MoPiX provides the user with a library of equations which use verbal terms
to identify the purpose of the equation, such as "greenColour(ME, t) = 100." This equation would
make object "ME", the object to which the equation is assigned, 100% green at time t. Since t
was not used in the equation in this instance, the color of the object would not have changed over
time and the object would have remained 100% green. Other library equations were provided
which allowed attributes to change over time. Two researchers, one a teacher at the school for
several years, observed, circulated among, and questioned students as they worked in groups of
two or three. The researchers also conducted whole class discussions. Students appeared to build
connections between the formal equations and the behavior of the objects. It seemed that the
equations became "tools for controlling and creating animated models", not just as they were
given in the library, but as refined or newly constructed by the students (Kynigos et al., 2010).
Exploring connections with graphing calculators and dynamic software. Hennessy,
Fung, and Scanlon (2001) examined adult students working with each other on activities
involving graphing calculators. Advantages they noted in the use of the calculators included
speed, visualization, seamless movement between representations, external reference to facilitate
discourse, helping to make thinking explicit, and encouragement to clarify ideas. The seamless
movement mediated the pairs? problem solving. The use of the calculator also encouraged them
75
to make their thinking explicit to each other and clarify their ideas. Disadvantages included the
need to assess mechanical vs. conceptual knowledge produced, and the researchers encouraged
the continued use of paper and pencil techniques to accompany the use of technology. Lopez
(2001) study, in discussing problems used as part of an algebra in-service workshop funded by
Casio, noted that used graphing calculators can be used ?as a visualization tool to make
connections between mathematical concepts? (p. 116). One such activity asked participants to
draw the ?golden arches? using functions. Another asked them to draw the outline of a stealth
bomber. By working such problems, questions about the graphs involved force participants to
?restructure their knowledge and connect it to the drawing that they are trying to display? (p.
118). Santos-Trigo (2002) used a series of tasks for which 25 first year university students taking
a course in calculus had graphing calculators and dynamic computer software available. Data
was gathered by means of student interviews and written reports. Three tasks were chosen to
demonstrate different features of student interaction with mathematics that emerged. The first
task involved examining a representation of quadratic equations as points in the plane, a
quadratic of the form y = x
2
+ bx + c being represented by the point (b, c). The visual approach
helped students understand the nature of the roots of such equations. This new type of
representation showed the importance of looking at mathematics from new perspectives. The
technology students used for the series of tasks assisted them in exploring connections among
different representations.
Yerushalmy and Sternberg (2001) used their Function Sketcher software to take seventh
grade students through three phases of learning algebra, graphic, iconic, and symbolic, in the
development of the concept of function. The software was able to receive mouse input and allow
students to draw on the xy plane. It also provided a ?stair step? view which could be added to the
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graph to show the rates of change for various changes in x and made available sections of graphs
with different characteristics, such as increasing slope, concavity, etc. It provided technological
tools which allowed students to experiment with objects used in representations of functions. The
continuous mouse input gave them a graphical representation. The graph sections tools were
iconic in nature and the stair step view gave rise to symbolic representations. Students were able
to see how to translate a real-life event into a graphical representation and how to use graphical
representations to build symbolic ones.
Lapp and John (2009) examined the ways in which pre-service teachers' mathematical
choices and conceptual understanding were affected by the use of "dynamically connected
representations" (p. 37). The technology used was a prototype of Texas Instruments' (2006) TI-
Nspire CAS
TM
. The pre-service teachers were able to observe patterns that they probably would
not have been able to see very easily without the use of technology. The researchers' hope was
that experiencing technology as learners would encourage the pre-service teachers to be more
likely to foster a student centered learning environment (Lapp & John, 2009).
Using technological laboratories to connect to real-life phenomenon. Microcomputer
based-laboratories (MBL), calculator based laboratories (CBL), and calculator based rangers
(CBR) allow the student a way to enter function information other than by equation. They also
allow for different types of visual analysis than conventional representations (Yerushalmy &
Shternberg, 2001). Students can study their own movement and discover relationships between
the associated numerical, graphical and symbolic representations (Stylianou et al., 2005).
Advantages of CBLs and MBLs include: multiple modalities, real events paired with their
symbolic representations, scientific experience, elimination of mathematical drudgery, and the
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encouragement of collaboration. Appropriate curriculum is needed in order for CBL and MBL
activities to be successful (Lapp & Cyrus, 2000)
The use of MBL activities was also examined by Hegedus and Kaput (2004), who
showed how their computer software, SimCalc MathWorlds could be used in a linked classroom
environment to promote discussion and engagement. The SimCalc project was based in part on
the idea that the phenomenon being graphed is itself a fourth representation (the first three being
equation, table, and graph.) An MBL could be used in conjunction with the MathWorlds
software as a connection to the phenomenon being graphed. Some students need physical action
as the fourth representation in order to understand the relationship of the graph to the real-world
phenomenon and to begin to move flexibly among representations (Stylianou et al., 2005). A
person's motion can serve as a semiotic embodiment when it is mathematical and facilitates the
understanding of mathematical symbolism. Re-enacting the motion becomes an executable
representation (Moreno-Armella et al., 2008).
SimCalc MathWorlds had the ability to take input from an MBL or to display an
animated image of a virtual actor moving in the way the graph described. Kaput (1998) referred
to the movement of the virtual actor as a ?cybernetic phenomenon? (p. 273). He felt that the
cybernetic or physical phenomenon should be at the center of the network of representations and
that the other representations be used to understand that phenomenon.
Lapp and Cyrus (2000), when observing high school students working with CBR?s at a
Mathematics, Physics, and Advanced Technology Exploration Day found that students did not
understand the graphical information during the activity, demonstrating common
misconceptions. ?To connect graphs with physical concepts, students need to see a variety of
graphs representing different physical events? (p. 3-4). They described advantages of CBL?s as
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including: multiple learning modalities addressed, real events paired with their symbolic
representations, scientific experience gained, elimination of mathematical drudgery, and
encouragement of collaboration.
Stylianou, Smith, and Kaput (2005) used CBRs with pre-service elementary teachers to
help them develop mathematical understanding. 28 preservice teachers attending a mathematics
course for elementary school teachers participated in a two week study in which they worked in
groups of four or five. CBRs were used in conjuction with calculators equipped with
MathWorlds software to allow the user to see the graph of the motion captured by the CBR and
to replay that motion (Stylianouet al., 2005). Questions were asked to determine the participants?
pre-existing understanding of graphs of motion. It was assumed that participants knew nothing
about CBRs or graphing calculators and they were introduced to both of them. Students were
asked to complete a task using the CBRs to collect data and represent mathematics in motion to
help them understand a position graph. Pre and post tests were given. The researchers found that
pre service teachers gained mathematical and pedagogical insights on graphs of functions when
working with the CBR devices (Stylianouet al., 2005). Mathematical insights included facing
their own misconceptions, realizing that graphs can be manipulated to allow for different views
and arguments, and using graphs as a means for mathematical communication. Pedagogical
insights included recognizing the value of building on students? kinesthetic experiences,
recognizing the need to link concrete experience to symbolic representations of that experience,
differentiating between local and global interpretation of graphs as tools for arguments and
recognizing the need to provide learning environments that allow for discussion and
communication about graphs (Stylianouet al., 2005).
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Linked whole group study. The SimCalc project's goal was to democratize access to
higher mathematical ideas (Hegedus & Kaput, 2004). Recent work added to their previous
studies a look at the potential of hand held wireless devices linked to larger computers. They saw
classroom connectivity (CC) as critical because of its potential to impact communication in the
everyday classroom even more so than internet connectivity. Technology can serve to be more
than just a medium for individuals, and become a "medium in which teaching and learning are
instantiated in the social space of the classroom" (Hegedus & Kaput, 2004, p. 130). They saw
this type of technology as aiding an epistemological shift in which technologically assisted
mathematical learning situations evolve from participatory simulations to joint constructions of
knowledge (Hegedus & Kaput, 2004).
Teaching experiments were conducted in middle and high school classrooms in
Massachusetts and California. The project described the grouping of the students by two
numbers (group number, count off number). The problem y = 2x + b, with "b" being the group
number was assigned to each group. Each student could produce the function on their own
device. The student's work was uploaded to the teacher, aggregated, chosen for display and
discussed. Graphs for students in the same group should have overlapped and their animated
objects should have moved alongside each other (Hegedus & Kaput, 2004). Organizing and
displaying student work helped focus attention on the underlying mathematical structure. When
using such an activity, before animating, asking what the race will look like will help students
think about what the mathematical representation is telling them (Hegedus & Kaput, 2004).
Triangulated data from pre and post test measures, video records, and field notes indicated that
participating students? algebraic thinking improved.
80
Assigning the students a group and count-off number and asking that those numbers be
incorporated into the function they are displaying gives students a personal connection to the
work, as they look to see how their construction fits in with those of other people (Hegedus &
Kaput, 2004). Combining dynamic representation with connectivity can help students understand
important algebraic concepts (Hegedus & Kaput, 2004). When dynamic mathematics software is
combined with digital networks, students' individual mathematical objects can interact in
meaningful ways. Students can share mathematical experiences (Moreno-Armella et al., 2008).
The use of dynamic geometry environments to deepen thinking. Dynamic computer
technology provides representations of mathematics which have not existed previously in the
external environment (Moreno-Armella et al., 2008). Cuoco and Goldenberg (1996) provided
examples of tasks from studies which had been done in both computational environments and
dynamic geometry environments which showed how students can build mathematical habits of
mind. Because computer environments perform exactly the instructions they are given, students
using them must think about essential mathematical features. One activity used a dynamic
geometry environment to examine a geometric construction to see how one segment changed
when other features of the construction was changed. The purpose of the example was to show
that technology can lead students to a style of thinking. Conjectures arose and were examined
further using the technology. The mathematical objects became real and became the subject of
experimentation. One concern was to determine whether or not the students were investigating
the mathematics or the properties of the software. They believed that when students were
engaged in the construction of the experiment, they would be more likely to feel that they were
experimenting directly with mathematical objects (Cuoco and Goldenberg, 1996). .
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Falcade, Laborde, and Mariotti (2007) used the trace tool in a dynamic geometry
environment to introduce students in four 10th grade classes (15-16 year old students, two
classes in France, two classes in Italy) to the concept of function by looking at covariation
qualitatively. Considering functions from a standpoint of covariation means that variations in the
independent variable (or input) and dependent variable (or output) are considered together. The
dragging tool and trace tool were used in conjunction so that the user could experience and
observe the combination of motions as an example of covariation. Students worked in pairs, but
wrote individually about what they had learned. The writings were done in a setting detached
from that in which the technological study took place. The students were also engaged in whole
class discussion, which was important to the process of making meaning of what had been done,
particularly in finding the search for a definition of function as it emerged from their
technological work. The teacher redirected the discussion to the main objective, prompted the
intervention of a student, repeated students? comments, pushed the discussion in important
mathematical directions based on student input, tried to involve non-participating students, and
orchestrated the formation of mathematical meaning using the contributions of students
(Falcadeet al., 2007). Evidence highlighted the importance of the teacher?s role in helping them
to make meaning from what they were doing. Students needed the help of teacher facilitated
discussion in moving from the technological experiences to a mathematical definition of function
based on those experiences, however the ideas on which the discussion were based emerged
from the students? work in the technological environment (Falcade et al., 2007).
This review has looked at several different ways that researchers have combined the
study of technology and representation in mathematics education. A theoretical framework for
the study will now be considered.
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Theoretical framework: Constructivism
I have chosen to approach this study using a theoretical framework of constructivism.
Constructivism emerged as a dominant theory in mathematics education in the 1980?s (Lambdin
& Walcott, 2007). Based on ideas of Piaget and Vygotsky, constructivism encouraged educators
to create an atmosphere where students could work through cognitive conflict using their own
strategies, and thus learning via problem solving (Lambdin & Walcott, 2007). Students then have
the opportunity to construct their own knowledge (Silver, 1990).
In addition to offering an account of student learning, constructivism is also an
epistemology and a research methodology (Ernest, 1998). Radical constructivism includes the
ideas that knowledge is not passively received and that all knowledge is constructed and reveals
nothing which can be applied with certainty to the world at large. Many researchers take issue
with the second idea, believing that we "inhabit a knowable external reality" (Ernest, 1998, p.
29). Social constructivism includes the idea that the social dimension of students' worlds affects
their learning, and that the knowledge constructed by the student is in response to socially
situated experiences (Ernest, 1998). Social constructivism includes Vygotsky's zone of proximal
development (ZPD), which refers to problems that a student may not be able to solve alone yet,
but that they can solve with just a little assistance, such as a facilitating question, or a hint
(Norton & D'Ambrosio, 2008). Norton and D'Ambrosio (2008) noted that Steffe defined a
different zone, the zone of potential construction (ZPC). He defined ZPC as the changes students
might make in their own understanding during or following mathematical interactions. The
teacher considers what he or she knows about the student's current way of thinking and considers
what might be changed about or added to that understanding. In addition to being a way of
looking at student learning, constructivism can also be seen in the work of the researcher who
83
constructs theory through the interpretation of data collected from subjects (Mills, Bonner, &
Francis, 2006).
Schwandt (2007) has said that in order to make sense of constructivism, it is important to
note what is being constructed. Constructivist frameworks allow me to make some sense of
students? interactions with technology. Creating an atmosphere were students can work through
cognitive conflict, observing the resulting changes students? make to their understanding, and
constructing theory through interpretation of data allow me to see and understand more about
their thinking and the internal representations they may be constructing. Such observations are
possible because mathematical representation is a dynamic, active process, in which students re-
present to themselves prior mathematical activities in ways crucial to the knowledge they are
constructing (Cifarelli, 1998). Monk (2003) described seeing as a constructed activity, closely
linked to thoughts and actions. Connell (1998) showed that students created personally
meaningful representations using computer based tools for solving problems.
Ernest (1998) noted that constructivist methods must be approached cautiously, with the
understanding that "there is no 'royal road' to knowledge" (p. 31). In addition, attention must be
given to beliefs, conceptions, language, and shared meanings of the subject and researcher
(Ernest, 1998). He contended that research methodology set in a constructivist epistemology be
conducted with caution and humility. Researchers may interpret the actions and language of
others but must remember that those others have their own realities. Qualitative researchers seek
to understand the realities of others in company with their own, acknowledging that such realities
can never be assumed to be fixed. The researcher is never external to the analysis (Ernest, 1998).
In keeping with these cautions, issues of validity were important to this study and will be
discussed in detail in chapter three.
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Conclusions and Questions
Constructivist frameworks are being used in this study to examine issues surrounding the
use of technology in mathematics education. Of particular concern are the internal
representations which adult developmental mathematics students may develop though
interactions with technology. There are both advantages and challenges to the use of technology
in mathematics education. With the rapid pace of its development, research is continually needed
in order to provide teachers of all ages with information to assist them in making wise choices
(Atan et al., 2008; Hollenbeck & Fey, 2009). The issues surrounding adult developmental
mathematics students are symptomatic of those challenges. Their presence in the educational
system points to gaps in their learning. The combination of academic and personal challenges
they face require adult developmental mathematics teachers to make careful choices about their
use of educational resources (Qi & Polianskaia, 2007). Figure 2 summarizes ideas found in the
literature and illustrates forces affecting adult learners. Technology has the potential to advance
student learning, but there are challenges to its use. Research is needed into how it can be used
beneficially.
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Figure 2: This diagram summarizes some of the issues adult learners face, benefits and
challenges related to the use of technology, and the question this study seeks to address.
Technology has a transforming influence on the role of representation in school
mathematics. Decisions about the choice of mathematics technology to use in the classroom are
strengthened by an understanding of the role of mathematical representations in student learning.
Technology can transform and add to the representations that are available to students. It can add
Benefits
Technology can help learners
visualize, work more quickly
and flexibly, learn to
communicate more
effectively, move more
flexibly among
representations, and increase
in confidence and satisfaction
Challenges
Technology requires
training, may result in
procedural learning,
may hamper valid
assessment, and
requires appropriate
tasks
Needs
Many adult learners need
remediation in mathematics.
Non-cognitive factors, low
literacy skills, lack of
experience in the educational
system, and lack of access to
trained faculty may affect their
learning.
Question
How can technology best
be used to address adult
learner?s needs and help
them build valid, useful,
and enduring internal
representations of
mathematics?
86
to a student?s ability to access, explore, analyze, and connect representations (National Council
of Teachers of Mathematics, 2000; Yerushalmy & Shternberg, 2001). For the internal
representations that students acquire through the use of technology to benefit them, those
representations must be valid, useful, and enduring. The following research questions form the
basis of the present study and address the issue of improving adult developmental mathematics
students? learning through the informed use of technology
1. Following the introductory use of dynamic computer technology to explore
mathematical concepts built upon previous knowledge, what internal
representations of those concepts do developmental mathematics students
possess?
2. What can be determined about the validity and usefulness of those
representations?
3. How well do those representations endure over a period of time and in the
company of tasks which build upon them?
This study was conducted to provide research-based evidence for developmental
educators by allowing inquiry based learning to take place in the presence of technology,
documenting students? thinking and learning in that setting, introducing students to accessible
resources, and focusing on mathematics which they are responsible for learning. Developmental
mathematics students need the increased opportunity that such a study provides. The following
chapter will provide details as to the specific methodology used.
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3. Methodology
This qualitative study was designed to gather knowledge about the effect of technological
representations on developmental mathematics students? understanding of functions. A sequence
of teaching interviews with developmental mathematics students was conducted, recorded,
transcribed, and analyzed. It was an exploratory case study, conducted with an eye toward
suggesting theory. Such theory generation does not require a large number of cases, since the
suggestion of new theory, rather than the proof or verification of existing theory, is the goal
(Glaser & Strauss, 1967). Further studies, some of which may take place over long periods of
time, can build upon the theoretical suggestions which have arisen. Glaser and Strauss (1967)
described theory as "an ever-developing entity" rather than "a perfected product" (p. 32). I chose
qualitative research as the most viable method for an in depth examination of student thinking,
and a teaching experiment as the most fruitful format in which such research could take place. I
also chose to examine the emerging data using ideas from grounded theory, so that unexpected
learning could be more readily and carefully examined. In order to provide a foundation for these
choices, a look at qualitative research, qualitative research in mathematics education, and
teaching experiments will begin the chapter. Grounded theory as an approach to data analysis
will then be examined. This is followed by a look at what was learned from a pilot study. A look
at the specific procedures used for the present study will follow, as will a look at my stance as a
researcher, and an examination of issues of reliability of validity.
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Theoretical foundations for qualitative research
It is helpful when considering the specifics of a study?s methodology to consider some of
the ideas from which those methods arise. A look at qualitative research in general will be
followed by an examination of qualitative research in mathematics education. The specific ideas
related to the conduct of teaching experiments as a form of qualitative research in mathematics
education will then be examined. A look at the relationship of case studies to teaching
experiments and tasks for teaching experiments will conclude this section.
Qualitative inquiry and foundations of thought. Qualitative data helps researchers
understand underlying relationships (Pandit, 1996). An observed situation is conceptualized,
specifics of the situation singled out within an analytical framework, a representational system is
used to analyze those aspects, the analysis is interpreted and inferences are made about the
original situation (Schoenfeld, 2007). It is an appropriate method to use when investigating
internal representations, such as mental imageries (Presmeg, 2006). Important contributors to the
field of qualitative research have been Eisner (1998) who discussed the idea of connoisseurship,
and Lincoln and Guba (1985) who described the nature of naturalistic inquiry. The term
qualitative has the advantage of encompassing many forms of human activity. It has also become
part of educational discourse. Rather than coining new terms, Eisner (1998) chose to refine the
discourse. As part of that refinement he referred to "qualitative inquiry", and indicated his belief
that the word inquiry had a broader application than the words "research" or "evaluation"
(Eisner, 1998, p. 6). His major focus was on educational connoisseurship and educational
criticism. Both types of analysis focus on qualities and though they are commonly considered in
relation to art, they can be applied to "educational phenomena" (p. 6). Connoisseurship requires
high levels of "qualitative intelligence" (p. 64). Our knowledge about a situation influences our
89
perception of it. When observing in a classroom for example, observers? perceptions would
change depending on their knowledge of the teacher's level of experience. They would think
differently of a first year teacher than they would of a veteran, though otherwise the situations in
which those teachers were working were the same. Knowledge about a situation can provide a
window, but it can also prove a hindrance. Labels and theories, for example, can promote
expectations which get in the way of perception. "[A] way of seeing is also a way of not seeing"
(Eisner, 1998, p. 67). For example, in order to combat that very idea, Edwards (1979) had her
drawing students turn the photograph they were copying upside down so that prior knowledge of
what they thought a subject should look like would not interfere with their current observations.
It is also helpful to consider the foundations of thought motivating methods of qualitative
inquiry. Lincoln and Guba (1985) described three historical eras of thought. The division of
those eras is centered in the emergence of positivism in the early nineteenth century, a movement
which can be characterized by the belief that the scientific method can be used to study diverse
topics and provide generalizable, exact, objective knowledge about those subjects. Before that
emergence, the prepositivist era was characterized by passive observation, rather than active
hands-on inquiry. In seeking to introduce the post-positivist era, characterized by a naturalist
paradigm, Lincoln and Guba (1985) described the emergence of non-Euclidean geometry. They
concluded, among other things, that "[d]ifferent axiom systems have different utilities depending
on the phenomena to which they are applied" (p. 36). Euclidean geometry and Non-euclidean
geometry are each preferred in different situations. In a similar way, a post-positivist or naturalist
paradigm allows the researcher to understand aspects of knowledge that positivism inadequately
addresses. Lincoln and Guba (1985) summarized the naturalist paradigm through five axioms,
which they contrasted with positivist ideas. Those five axioms include the following ideas. There
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are multiple, holistic realities. The knower is inseparable from the known. Only idiographic (case
specific) statements are possible, rather than universal generalizations. It is impossible to sort
causes from effects because of mutual interdependence, and research is bound by values,
including those of the researcher, the theories they employ, and the context within which they
work (Lincoln & Guba, 1985). The present study will assume a post-positivist paradigm, noting
that the research being conducted here is case specific, affected by the views and values of the
researcher, and limited by the knowledge the researcher brought to the study regarding existing
theories applicable to the research.
Qualitative research in mathematics education. Quantitative research methods have
occasionally demonstrated some important relationships in mathematics education, but have
rarely explained those relationships. Mathematics education researchers have turned increasingly
to qualitative research (Silver & Herbst, 2007). The increase in qualitative research in
mathematics education has raised the demand for theory on which to base such research. Theory
can mediate the bidirectional relationships between problems, research and practices (Silver &
Herbst, 2007). It mediates between problems and research by giving meaning to results of
research studies, providing a lens with which to look at data, or providing a tool to describe a
body of research. It mediates between research and practice by prescribing what educational
practices should be like, helping researchers understand observed practices, providing language
to describe practices, explaining causes for practices, and predicting aspects of practice. It
mediates between practice and problems by providing a proposed solution to a problem, by
establishing criteria by which different instances of problems can be compared, by identifying
different types of problems, by identifying aspects of practice that pose or contribute to
information about problems, by helping design new practices, and by helping justify choices in
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addressing problems (Silver & Herbst, 2007). Any suggested theory which may arise from the
present study will be considered in light of such mediation. The specific approach taken in the
present search for theory is a teaching experiment.
Specific approach: Teaching experiment. The decision as to the type of approach to be
used in research is influenced by the goals of that research (Creswell, 2007). The goals of
research in this case are to examine the effects of technological representations on developmental
mathematics students? internal representations of functions. This requires the introduction of
subjects to a technological mathematics environment and a deeply developed examination of
their thinking. A teaching experiment provides a way to both introduce a teaching tool and
examine student thinking. Researchers have used teaching experiments to investigate student
thinking (Steele, 2008). As Lesh and Kelly (2000) have stated, teaching experiments can be used
when it is desired that insight be gained into processes that bring student thinking from one state
of knowledge to another. They allow conditions to be created which optimize the chance that
change will occur while leaving open the direction in which the student?s knowledge can develop
(Lesh & Kelly, 2000). In their work examining the influence of technology, Falcade, Laborde,
and Mariotti (2007) also noted that a teaching experiment is appropriate for introducing students
to a mathematical concept.
Exploring student thinking. The ideas of valid, useful, and enduring internal
representations developed from a review of the literature for this research study and were carried
into the study as part of the a priori coding of events which would transpire there. A teaching
experiment would be a fitting setting in which to explore such ideas. Enduring representations
are those which stay with the student in various situations, are carried over to later work,
deepened and built upon as the student progresses mathematically. The teaching experiment can
92
be helpful in examining enduring representations, since it adds to the clinical interview by
examining progress made over a period of time as opposed to just looking at the subject?s current
knowledge (Steffe & Thompson, 2000).
Valid representations have been defined by the researcher as those that accurately
represent the mathematics they seek to represent. Useful representations are those which can be
selected and applied to maximize problem solving, build new knowledge, and communicate
mathematical ideas to others. A teaching experiment is useful for examining these two
constructs, because of the deep exploration of the subject?s thoughts and the development of
those thoughts in various circumstances (Nemirovsky & Noble, 1997; Steffe & Thompson,
2000).
As the teacher-researcher observes the subject in a teaching experiment, attempts on the
part of the student to resolve a mathematical problem may confirm the student's mathematical
reality, showing how they think (Steffe & Thompson, 2000). This ability of a teaching
experiment to enlighten the researcher as to student thinking also makes it a viable method for
examining the impact of technology on a student?s internal representations for mathematics,
since, by definition, those representations exist in the student?s mind (Goldin, 2003).
Nemirovsky and Noble (1997) noted how the subject?s thoughts developed, how she ?came to
recognize? certain mathematical behavior ?by visual inspection? (p. 99). In their teaching
experiment, in which they interviewed a subject for three one-hour sessions, they also noted her
efforts to organize her thoughts as related to the visual experience with a computer based-tool
which created graphs of height vs. distance and slope vs. distance (Nemirovsky & Noble, 1997).
Their study demonstrates that teaching experiments allow the examination of teaching tools and
the development of student knowledge within the environment that tool provides (Lesh & Kelly,
93
2000; Nemirovsky & Noble, 1997). The emphasis in teaching experiments on examining student
thinking can also be seen in the selection of subjects for such a study, since in some cases
students with particular thinking patterns were selected (Norton, 2008).
One way of looking at student thinking is to find out the schemas they develop for
solving problems (Steele & Johanning, 2004). A schema is ?a mechanism in human memory that
allows for the storage, synthesis, generalization, and retrieval of similar experiences? (p. 66).
Steele and Johanning (2004) cited Piaget as using the term ?cognitive structure? (p. 66). It lets
the learner recognize similar experiences. Abstraction and reflection are two vital mental
processes in the development of schema. An experience may provide a memory upon which a
schema can be built. That mental construction may be triggered when a new situation arises for
the learner to process. Schema are not memorized in the traditional sense of memorizing a
formula, but are built up, becoming deeper and more connected to other knowledge (Steele &
Johanning, 2004). Steele and Johanning (2004) noted that one strength of a teaching experiment
is the opportunity to engage in conceptual analysis, looking "behind what students say and do"
(p. 70). Teaching experiments allow researchers to experience students' learning and reasoning
firsthand. They test as well as generate hypotheses. The basic goal of a teaching experiment is to
examine what students say and do while engaged in mathematical pursuits and to model students'
mathematics (Steffe & Thompson, 2000). Part of understanding the students? mathematics is
understanding what they cannot do or understand. This also emerges in the teaching experiment.
The researcher can then consider what rationality lies behind the students? choices (Steffe and
Thompson, 2000).
Case studies and teaching experiments. Suter (2005) noted that over 50% of researchers
surveyed had used the case study method for their research. Educational case studies may
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involve teaching experiments, but they may also use other techniques, such as a focus on
understanding students? existing knowledge or the effect of implementing a certain instructional
approach.
What students do versus what they might do. Sometimes the researcher's aim is clearly
stated as the understanding of existing student thinking, as in Sajka's (2003) work in which the
object of the study was to examine an average student's understanding of function. The
researcher in this case chose non-standard tasks which required the student to look at functions
from a different perspective than the one to which they were probably accustomed. Sajka (2003)
stated, "I was not interested in the pupil's ability to solve the problem on his or her own, but
rather in observing the process of finding a solution" (p. 232). By observing problems the
student encountered, the researcher gained information about the student's understanding. The
student in this study had been learning about functions for three years, knew the formal
definition, and was familiar with examples and representations (Sajka, 2003). The report was
limited to an analysis of one dialogue which dealt with this task: "Give an example of a function
f such that for any real numbers x, y in the domain of f the following equation holds: f(x+y) =
f(x)+ f(y)" (Sajka, 2003, p. 233). The questions used by the researcher during the course of the
interview helped clarify the student's understanding. The resulting dialog provided the researcher
with information about the student?s understanding of function and its associated symbolism.
Sajka (2003) was able to conclude that the subject's concept of function was not a "fully-fledged
mathematical object" (p. 252). The choice of a task as one that was not conventional was a key
factor, as it pointed also to "the influence of the typical nature of school tasks leading to standard
procedures" (Sajka, 2003p. 253). In this way, the researcher, seeking to examine student
thinking, also shed light upon an aspect of teaching.
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Though examining student thinking occurs in this case, a case study investigation into
how a student does think is different in its goals and activities from a teaching experiment which
examines the way a student may think. Steffe (1991) stated that "[t]he constructivist teaching
experiment is a technique that was designed to investigate children's mathematical knowledge
and how it might be learned in the context of mathematics teaching" (Steffe, 1991, p. 177). The
role of the researcher in a teaching experiment is more than that of an observer. The researcher
becomes ?an actor? constructing models of what is occurring in the student?s mind as a result of
the researcher?s actions (Steffe, 1991, p. 177).
The choice to use a framework which considers all that students may think in regards to
their experiences with technology permits the researcher to follow leads which may occur and
turn the project into a truly exploratory case study from which possible theory may emerge. The
possibilities for such theory are thus expanded. In a teaching experiment, as opposed to classic
design, interactions that use the researcher?s mathematical knowledge are allowed, and student
sense-making may emerge (Steffe & Thompson, 2000). The "researcher acts as teacher" in an
"interactive communication" with the subject with the goal of finding out what the subject may
learn and what may foster that learning (Steffe, 1991, p. 177). Steffe (1991) described the
teaching experiment as "an exploratory tool . . . aimed at investigating what might go on in
children's heads" (p. 177).
Descriptions versus adjustments. In addition to adding to standard examinations of
student knowledge by exploring what they may be able to learn, the exploratory and flexible
nature of a teaching experiment also adds to standard explorations of instructional approaches in
that the researcher's continued actions are based on the subject's actions. There is no
predetermined way of solving the problem presented. The researcher bears the responsibility of
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making "on-the-spot" decisions based on what is happening in the experiment and the emerging
model of the student?s knowledge (Steffe, 1991, p. 177). In one teaching experiment, 6 pairs of
children worked with a teacher/researcher for about 45 min. per week for about 75 weeks over a
period of 3 years, outside their classroom, using computer tools which allowed the students to
use discrete sets of objects and continuous line segments to model fraction concepts (Olive,
1999). The introduction of technology is an instructional approach and certain pedagogical
approaches may be present with the use of technology that may not otherwise have been there
(Kaput & Thompson, 1994). One of the tasks discussed in the study by Olive (1999) was to
consider sharing part of a pizza among friends and find out how much of a whole pizza each
friend would receive. The researchers made hypotheses during the experiment when they saw a
difficulty a student was having. They introduced a constraint that might move the student
forward by refocusing his attention on an overlooked aspect of the problem. The instructional
approach was adjusted during the course of the experiment. The teacher?s questions guided the
student?s thoughts (Olive, 1999).
Contrast this approach to a case study done by Butler, Beckingham, and Lauscher (2005)
which looked at "the processes by which . . . students were supported to self-regulate their
learning in mathematics more effectively" (p. 156). The instructional model used was Strategic
Content Learning (SCL). Four SCL principles of instruction were described: "Integrate support
for self-regulation into instruction . . . . Students as active interpreters . . . . Learning in
mathematics as guided (re)construction" and "Learning in pursuit of a goal" (Butler,
Beckingham, & Lauscher, 2005, p. 160). A formal evaluation of SCL had previously been done.
This study looked at three eighth-grade students and asked how SCL achieved instructional
goals, how the students' learning was mediated, and how SCL was used in responding to
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individual needs. Individualized Educational Plans (IEPs), psychoeducational assessments, and a
Metacognitive Questionnaire gave the researchers information about the students (Butler et al.,
2005). The researchers also conducted observations, collected teachers? written reflections,
analyzed videotapes of instruction, and looked at student work samples and strategy sheets. They
kept track of test performance, including an analysis of data from examinations given early and
late in the project to provide pretest and posttest information. Their interpretation of data
provided "a descriptive account of SCL intervention in relation to student learning" (Butler et al.,
2005, p. 162). Researchers created narratives described how SCL worked for each case. They
also completed a cross-case comparison.
Multiple data sources were used in the study by Butleret al. (2005) and a particular
instructional model was examined, the work done was observational and descriptive of what was
happening between the instructor and the student. The SCL model provided built in
encouragement to adjust to student needs, and the teacher observed did so. The study provided a
?rich description of instructional processes? (p. 172). The study by Butleret al. (2005) examined
an instructional process and provided a description of it. The Olive (1999) study tried an
instructional approach and adjusted it based on researcher observations during the course of the
implementation. This quality of adjusting an approach based on researcher observations during
the course of a study is characteristic of the teaching experiment, in which, as has been noted, the
researcher is ?an actor making ?on-the-spot? decisions in order to maximize the exploration?
(Steffe, 1991, p. 177). This potential for adjustment to the teaching approach based on
observations during the data collection also makes a teaching experiment more suitable for the
current research study than a traditional examination of a teaching method, since, as has been
noted, the study is an exploratory one, seeking to suggest theory. When potential theory
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development is a goal, the researcher must be allowed the freedom to follow such leads as may
arise and question the subjects enough to develop the ideas that are emerging (Creswell, 2007).
This pursuit of emerging ideas is consistent with the adjustable nature of the teaching experiment
which has been described. A merely descriptive study would not provide sufficient flexibility to
the researcher.
Tasks for teaching experiments. Rather than guiding the students toward a definite
answer, teaching experiments present students with tasks that invoke in them a need to develop
new interpretations or refine their thinking. The tasks can allow students to learn and to
document their learning through built in descriptions and explanations (Lesh & Kelly, 2000).
The tasks should allow the researcher to ?bring forth and sustain students? independent
mathematical activity? (Steffe & Thompson, 2000, p. 293). The questions accompanying the
tasks should elicit conjectures based on the research questions or hypothesis (Norton, 2008).
To see how tasks for knowledge assessment and for assessing instructional approaches
might differ from tasks used in a teaching experiment, consider the following task:
Describe the effect that changes to m and b have on the graph of the equation
y = mx + b.
If the goal was to assess the student?s knowledge about this topic, the task could be given
to them as is, with no tools other than the knowledge possessed by the student and no prior
interventions. If the researcher?s goal was to test an instructional approach, the instructional
approach would need to be implemented prior to administering the above task. For purposes of
this discussion, suppose the instructional approach is the use of technology to teach this topic.
The same task might be given to different groups, one group receiving it after the students have
received instruction in the use of the technology in question, for example dynamic sketches in
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which they explored how changes in the parameters affected the graphs. The researcher might
then administer the task and note the effect of the technological instruction on the subjects?
knowledge. In contrast, a teaching experiment task should
? Provoke the student to develop and refine their thinking (Lesh & Kelly, 2000)
? Provide built in opportunities for them to describe and explain what they are
doing (Lesh & Kelly, 2000)
? Allow the researcher to bring out mathematical activity in the student (Steffe
& Thompson, 2000)
? Allow the researcher to follow up on research questions and hypotheses
including those that arise during the course of the data collection (Norton,
2008)
Summarizing these ideas for practice, the task for a teaching experiment must be thought
provoking, built on prior knowledge (so that it is accessible and thinking may be refined), open
ended, provide prompts which encourage description and explanation, and be accompanied by a
semi-structured interview protocol which is flexible enough to allow the researcher to make the
necessary explorations for their study. Such ideas were taken into consideration in developing
tasks for the current study.
Steffe (1991) has said, ?a general goal of mathematics teaching is for teachers as well as
students to learn, and the primary goal of the constructivist teaching experiment is but a
microcosm of this general goal of mathematics teaching" (p. 192). The key quality indicated here
is that the researcher?s and subject?s learning are intertwined and occurring simultaneously. This
type of research will allow the investigator to examine the effect that the introduction of
technological representations to the student has on their thinking (Falcade et al., 2007). The
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range of possible effects that might be discovered is widened in a teaching experiment with its
flexible and exploratory nature over case studies which merely look at student?s prior knowledge
or the effect of a particular instructional approach. The teaching experiment looks at the
student?s knowledge as it develops and allows the adjustment of the instructional approach to
permit the investigation of emerging theory (Steffe & Thompson, 2000). Teaching experiments
generate detailed data in the interactions they track. Following is a discussion of theory which
will help formulate a method for analyzing the data that has been generated.
Approach to data analysis: Grounded theory. The examination of data resulting from
the teaching experiment conducted in the present study is inspired by the pattern described by
Glaser and Strauss (1967) in their classic description of grounded theory. A researcher engaged
in this method of analysis would first code his data into as many categories as are possible,
taking care to compare different pieces of data in the same category. Theoretical properties of the
category emerge from this comparison. As this coding continues, the researcher can periodically
stop coding and use memoing to record thoughts and theoretical ideas (Glaser & Strauss, 1967).
During these periods of reflection, care should be taken that any logic is grounded in the data and
not in speculation. As more is learned about categories, different categories become integrated
with each other when relationships between them become apparent. Questions may also emerge
which may guide the subsequent collection of data. Glaser and Strauss (1967) noted that this
continued coding and integration process leads to the reduction of categories to a "smaller set of
higher level concepts" (p. 110). As this process moves forward, it is affected by what Strauss and
Corbin (1990) referred to ?theoretical sensitivity? which they described as the researcher?s
ability to notice the meaning in data.
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They noted that a theoretically sensitive researcher will be able to separate important
from unimportant qualities in the data. The ability to do so allows the researcher to formulate
well-grounded theory more quickly than he or she otherwise would and give the data meaning
which is "faithful to the reality" (Strauss & Corbin, 1990, p. 46). Their ideas are in keeping with
Eisner?s (1998) notions of connoisseurship in qualitative inquiry. Theoretical sensitivity can be
acquired by a study of the literature, through professional experiences in a particular field,
through personal experiences related to the field of study, and as a by-product of the analytic
process (Strauss & Corbin, 1990). The comparisons and ideas arising during analysis lead to
other ideas which may result in a closer look at previously examined data and the discovery of
new meanings.
Lincoln and Guba (1985) looked at the data processing aspects of Glaser and Strauss?s
(1967) constant comparative method (p. 340). They expanded the information related to stage
one of the process, "comparing incidents applicable to each category", noting that the emergence
of categories involves more "effort, ingenuity, and creativity" than the statement that "categories
'emerge'" might imply (p. 340). Semantic relationships, for example, might be difficult to
identify. A relationship might be inclusive, and be described as "x is a kind of y", or it might be
sequential, allowing it to be referred to as "x is step (stage) in y" (p. 340). They stressed the
importance of the analyst?s ''tacit knowledge" as used on the first pass through the data, in which
assignments to categories may be made based on what seems right to the analyst. Later analysis
can clarify such understanding, but Lincoln and Guba (1985) claimed that such impressions may
be hard to capture later. It is important that new incidents assigned to a category be compared
with previous entries in the category. Thinking about such things assists in the process of
refining categories. When conflicts arise, memos can help the analyst capture his or her thoughts,
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providing an outlet for any conflict he or she feels and helping him or her discover the category?s
properties. Stage two of the analysis finds the analyst shifting to a more rule-based system of
classifying incidents. Subcategories may need to be formed or categories may need to be
redefined. The analysis then comes closer to describing what is being studied. Lincoln and Guba
(1985) noted that "fewer and fewer modifications will be required as more and more data are
processed" and that as categories become clearer "options need no longer be held open" (p. 343).
They said that "categories become saturated, that is, so well defined that there is no point in
adding further exemplars to them" (p. 343-344). Saturation may also be described as that point at
which "continuing data collection produces tiny increments of new information in comparison to
the effort expended to get them" (Lincoln & Guba, 1985, p. 350). Specific methods for applying
grounded theory techniques to the present teaching experiment will be discussed following the
summary below.
Concluding ideas about methodological theory. This study is based in the ideas of
constructivism. It assumes that both researcher and subject construct knowledge together during
the course of the particular methodological approach taken here, the teaching experiment.
Qualitative research methods allow the researcher to collect and examine data with the eye of a
connoisseur and use the rigorous data analysis methods of grounded theory to construct ideas
which are both truthful and important. The next portion of this chapter will describe some of
what I learned from conducting a pilot study. The specific procedures which were used in the
final study are then described. This will be followed by a look at my stance as a researcher, and
issues of reliability and validity.
Methodology learned through the pilot study. Since I had not previously conducted a
teaching experiment or tried the video techniques that would be used for this study, I conducted a
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pilot study. I recruited Steve
4
, a male in his early 20?s, who had been out of school for about 4
years and did not remember taking Algebra 2. He participated in 5 sessions for a total time of
about 3 hours and 34 minutes.
Facilitating Steve?s work gave me valuable experience in the type of progress adult
developmental mathematics students might be able to make and the kind of questioning and
facilitation it might require. I also gained practice in basing my actions on his actions and
investigating the potential of his thinking, as described by Steffe (1991), rather than trying to
guide Steve to a particular goal I had in mind. As I did this I found that Steve ?came alive? when
he was allowed to pursue his own ideas. One goal of a teaching experiment is to ?bring forth and
sustain students? independent mathematical activity? (Steffe & Thompson, 2000, p. 293).
In addition to gaining experience allowing him to work at his own pace, I learned more
about providing the type of facilitation that is sometimes necessary to focus the subject's
attention on pertinent mathematical relationships which will allow them to use mathematical
ideas they already possess in new situations Olive's (1999). This occurred when I encouraged
him to graph a vertical line so that he could visually follow the relationships of the x value to the
point on the function he had graphed and the related y value more closely. As I worked with
Steve I also found that the interview protocols I started with were too complex to use or follow
well in a teaching experiment setting and was able to simplify them considerably for the final
study.
A brief summary of the pilot study results, including the presentation of selected video
clips from the pilot study, was presented to a group composed of two mathematics educators, a
specialist in educational research, and a mathematician. Technical suggestions were made
4
All names are pseudonyms
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regarding the video set-up so that the appearance of the final product would be more easily seen
and understood. Ideas for strengthening the study?s theoretical investigation were also suggested.
Particular attention was drawn to the nature of a teaching experiment and the perceived need to
focus not on eliciting student understanding but on finding out how they come to understand.
One technique which was clarified through the conduct of a pilot study was the practice
of taking into account what happened previously in preparation for each new interview session.
Some things that might have been investigated were to note more fully the role of technology in
enhancing the usefulness of the internal representations that Steve possessed. Another possible
idea was to see how much the things Steve determined from his visual examination of the graphs
and his examination of how they changed translated into increased algebraic understanding.
Though such intriguing questions may arise in the researcher?s mind, the pursuit of knowledge
must be balanced with the pace at which the subject can be made to reveal information without
imposing upon him or her the researcher?s own thinking. A pilot study such as the one described
here can be an invaluable opportunity, particularly for new researchers, to learn more about that
balance.
Procedure
This discussion of specific techniques used for this study includes information about the
selection of cases, instrumentation and data collection, data analysis, stance of the researcher,
and issues of reliability and validity. It will then be followed by a brief examination of the pilot
study which was conducted in preparation for the final teaching experiment.
Selection of Subjects. The following section starts with a look at how an institution
appropriate to the needs of the study was selected. Following that, the process used to recruit the
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subjects of this study as well as the procedures used to decide the ordering of the work done with
the subjects will be described.
Institutional selection. Subjects were recruited from and the data collection conducted at
Harrisville State University
5
(HSU), a mid-sized university in the southern United States offering
both undergraduate and advanced degrees. This institution was selected because of its proximity
to my home and the presence of a large adult developmental mathematics population from which
it was hoped that a sufficient number of subjects could be recruited. HSU has provided
substantial support to returning students and others from the local community. That support has
included a college within the university offering developmental courses, including two course
offerings in mathematics
6
, Math 98 and Math 99. Three faculty members at the college
specialized in those developmental mathematics courses. In addition, a preparatory algebra
course, Math 100, was also available to assist those students whose mathematics placement
scores indicated the need for remediation, but who had not been required to test for learning
support. This course was also considered to be at the developmental level, since it did not count
for credit toward graduation. An inspection of course offerings during a recent spring semester
including the two developmental mathematics courses and the preparatory course showed that 26
sections of mathematics at the developmental level were being offered, with a population of
almost 600 students, and employing at least 15 different faculty members. This attention to the
developmental mathematics population and the resulting presence of large pool of potential
subjects in close proximity to my own location made this an ideal setting for the recruitment. The
5
pseudonym
6
Course designations have been changed
106
cooperation of personnel within the university in providing space and allowing recruitment also
added to the benefits of doing the research at HSU.
Case selection. Both passive and active recruitment methods were used to find potential
subjects for this study. With approval, the researcher entered developmental and preparatory
course classrooms at the study site, and delivered a brief invitation to participate in the study.
Flyers were provided at that time to potential participants, describing the study and including
contact information. Flyers were posted in areas nearby those classrooms. Since this was an
exploratory case study, two to three cases were deemed to be sufficient (Creswell, 2007). Five
people responded to the recruitment to the extent of providing personal information and
indicating an interest in the study. On one occasion I had appointments set up with three subjects
in the same day. One of the subjects got lost, but called and was able to make the appointment.
The other two did not show for their appointments. I was eventually able to get three initial
interviews with three different subjects. Even though I had intended to select from a pool of up to
8 initial interview participants, the response was lower than expected. The three who did
respond, however, were each enrolled in a different developmental mathematics course. Their
initial interviews confirmed the notion that they represented a spectrum of mathematical
experiences, and I decided to stop recruitment and use those three subjects. Recruitment and data
collection took place during Fall Semester 2009.
The three subjects were Shirley, Marlon, and Marjorie
7
. Shirley was a 46 year old
African American female enrolled in Math 98 who had been enrolled in the same course the
previous Spring, but had not completed it. Previous mathematics courses she listed included
general mathematics and Algebra 1. Marlon, a 53 year old African American male, was enrolled
7
All names are pseudonyms
107
in Math 99 and listed Math 98 as a previous mathematics course. Marjorie, a 36 year old African
American female, was enrolled in Math 100 and could not remember a previous mathematics
course, indicating that it had been 14 years since she had been in high school.
Miles and Huberman (1994) have noted that with a small number of cases, purposeful
sampling can allow the sample to be chosen to fit the logic of the study. Random sampling would
be less effective for the purpose of the study. Part of the purposefulness is in the setting of
specific boundaries for the section of cases (Miles & Huberman, 1994). One boundary for this
study is enrollment in developmental or preparatory mathematics at HSU. The cases selected
provided a range of mathematical experiences, providing different perspectives for the study
(Creswell, 2007). Although all of the subjects were African-Americans over age 35, they each
showed different mathematical understanding in their initial interviews. They were also enrolled
in different level courses of study for developmental mathematics students. This shows that
although there were only two subjects, these two subjects represented different portions of the
developmental mathematics population.
Since the idea of the study was to examine the influence of technology on the student?s
internal representations of mathematics, the previous experience of the potential subjects with
technology in their mathematics learning was also one of the areas I examined through the initial
interview. Both Shirley and Marlon indicated their use of the internet, particularly a computer
aided instructional program used by their mathematics teacher. Marjorie indicated experience
with calculators. Logically, the intervention used in the teaching interviews would have greater
impact on inexperienced students? internal representations than it would for those who have
previously internalized technological representations. Though they had some experience with
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technology, none of the subjects indicated any experience with the software to be used in the
study, Geometer?s Sketchpad. This added to their suitability as study subjects.
Marlon was selected to be the first participant in the study and Shirley and Marjorie were
notified that they would be contacted later in the semester following my work with him. Since he
was enrolled in the second developmental mathematics course, this would presumably allow a
more central set of data to be collected and the two cases representing either extreme of the
population could then be compared with his data. Unfortunately, due to illness and other
challenges, Shirley was only able to attend the initial interview and one other session. Attempts
were made to contact her early the following semester, but she did not make it to a follow up
interview, and eventually stopped returning phone calls. Since two cases were deemed sufficient,
the study was completed with Marlon?s and Marjorie?s cases.
Instrumentation and data collection. Interviews were the source of data for this study
and the mode and structure of those interviews is described below. A discussion of the general
interview technique is followed by specific looks at the initial interview and the series of
teaching interviews which followed it.
Interview technique. Interview protocols are found in Appendix B. All interviews, both
the initial interviews and the teaching experiment sequence, were semi-structured and in
accordance with effective interview techniques, efforts were made at the beginning to put the
subject at ease, gradually building to more direct examinations (Kvale, 1996). I designed
questions which encouraged the subject to relax, to use language comfortable for them, to be
descriptive, and to be specific about their experiences. At times I rephrased and repeated
questions and encouraged them to explain what they were thinking to increase the validity of my
interpretation of their responses (Kvale, 1996). Some brief field notes were collected during the
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interviews, but these were minimal as my attention was focused on conducting the teaching
experiment and responding to circumstances. I kept more detailed notes in a descriptive word
processed journal in which I recorded my reactions to sessions. Any reference to the subject in
notes, journals, and transcriptions was made using a pseudonym.
Initial interview. Initial interviews can help researchers gain understanding of prior
knowledge so that in addition to gaining other information for subject selection, they will gain
knowledge about the subjects? experiences with similar mathematical tasks (Hollebrands, 2004).
The initial interviews held in the present study lasted from about 35 to 45 minutes. Following
appropriate measures to ensure informed consent of the subject, I collected data regarding the
subject?s demographics, times of availability, and educational background. Any reference to the
subjects? real names was kept in a locked file box along with copies of their signed informed
consent documents. I collected data regarding mathematical background by listening to the
subject?s anecdotal accounts of their educational experiences. I allowed them to share
mathematics of their choice which they remembered, and presented them with a diagnostic task.
Initial interview subjects were informed that they may or may not be selected for further
participation. Note that provision was made for additional questions other than those listed on the
protocol form.
Teaching experiment interviews and tasks. "Learning how to bring forth and sustain
students' independent mathematical activity is a part of learning how to interact with students in
a teaching experiment" (Steffe & Thompson, 2000, p. 293). Koichu and Harel (2007) suggested
that interviewers encourage the interviewee in his or her thinking, that they not probe deeply
early in the interview, and that they engage in a semi-structured conversation. With these things
in mind, a sequence of tasks was designed to be presented to individual subjects. They were
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created to investigate and advance the subjects? understanding of different representations of
functions at a level consistent with their exhibited prior knowledge. However, not all of the
tasks prepared for the study were actually used. The tasks as designed can be seen in Appendix
C.
The first tasks, designed to be used in the initial interview, consisted of the examination
of patterns. After the subjects had been permitted to share some mathematics of their choice,
they were presented with the task ?Looking at patterns? and examined the sequence of shapes.
They were then presented with the task ?Looking at dot patterns? to see what patterns they could
find there. Should it be needed, a third task, ?Soda Cans? was also prepared, but it was not used
in the actual study. The task ?Another dot pattern? was used as a supplement to ?Looking at dot
patterns? and arose from work done in the pilot study. A sequence of tasks situated in
technological settings was also prepared. The technological tasks were designed to allow both
written and technological representations. The tasks were adaptable to the specific needs of the
subject, allowing multiple entry levels, and open-ended responses reflecting student thinking.
They were also designed to elicit information about the subjects? understanding of ideas and
representations associated with functions. Probing and specifying questions were used as needed
to elicit student thinking, as supplements to the protocol, consistent with semi-structured
interviews.
Technological procedures. The mathematics software selected for the study was
Geometer?s Sketchpad v. 4.07s (Key Curriculum Press, 2006), which incorporates graphical,
tabular, animated, and symbolic representations. This software was installed on a guest account
on my laptop computer which I took to each session. I also provided a movable mouse for the
subjects so they would not be hampered by the use of the laptop?s touchpad mouse.
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There were other technological considerations as well. Because the study was to be a
teaching experiment, a record was needed of all of the pertinent actions and statements made so
that they could then be analyzed. These actions included the subject?s interactions with the
software, the subject?s creation of representations on paper, and the physical actions of the
subject and interviewer. In order to capture all of these interactions, and at the suggestion of my
advisor, three simultaneous recordings were made and those recordings later synchronized into
one video production. Campbell (2003) described the use of "dynamic tracking . . . to capture a
complete record of a learner's interactions with a [computer based learning environment] in real
time" (p. 73). In order to capture the learner's interactions as completely as possible, Campbell
(2003) made simultaneous video recordings of the learner and the learner's computer screen. The
work done in the present study adds to these two the work done on paper as well. The
synchronization required that either the paper view or physical movements view be synchronized
first as a picture in picture (PIP) with the screen capture view and then the other view added as a
PIP to the resulting video.
The three recordings were produced as follows. One recording was captured by software
which was used to create a video record of the activities on the computer screen. A small camera
was placed on a small tripod which sat on the table and faced downward to capture what was
happening on paper. A blank piece of paper was taped to the table so the subject knew where to
keep their written work so it would be in view of the small camera. Another camera was set up
across the room on the other side of the table from the subject to capture the physical
movements. Figure 3 is a still capture from one of the final synchronized videos.
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Figure 3. Still capture illustrating the camera set up and assembled video recordings
In the third session held with Marjorie, a technical error resulted in the loss of the
recording of the computer screen, however sufficient data from the other two feeds provided
insight into what happened during that session and implications were still possible through the
conversations and paper representations which were preserved. A total of 7 sessions were held
with Marlon for a total time of about 6 hours and 47 minutes. A total of 6 sessions were held
with Marjorie for a total time of about 5 hours and 3 minutes. An examination of how the
captured data was analyzed follows.
Data analysis. It is essential that those engaging in teaching experiments plan adequately
for the labor-intensive activity of retrospective analysis, including a careful examination of the
videotapes (Steffe & Thompson, 2000). In this case, videotapes included a record of the
computer screen, the subject's reactions, and the subject?s written work, necessitating extra care
in observations. As noted, video recordings were synchronized via computer software, so that
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interactions between researcher and subject, computer screen action, and the subject?s written
work were viewable simultaneously. Care was taken to note mouse movements along with
transcription of the spoken word, and occasional notation was also included as to the timing of
episodes within the session being transcribed. Transcription technique was inspired by Campbell
(2003) and includes actions within braces {} and timing within brackets []. For example, the
following transcription shows that Marjorie had the mouse pointing at the point (10, 6) on the
xy-plane after she said the word ?itself?, moved it to (0, 5) before finishing her next statement,
and then moved it so that it was at (0,0) before she said the word ?So.? It also shows that this
statement ended at about 37 minutes and 56 seconds into the rendered video recording.
MARJORIE: It?s got to intersect with the graph itself {cursor now at (10, 6)} because it
does not have any, um {cursor now at (0,5)} ? its like to the exact ? it is rounded up or
its like rounded {cursor at (0,0)}. So it doesn?t go to like any of the, the cents [37:56].
Following is a look at the coding techniques used in this study including a discussion of
how emergent codes were addressed. One important emergent code is described.
Coding techniques. A unitizing and coding guide is provided in Appendix D for the
reader?s convenience and may be referred to during this discussion. This guide includes
definitions for unitization, and a listing of families of codes with a priori and emergent codes
noted as is appropriate within each family. Definitions and examples are given for each code.
Initial coding was done during the course of the data collection, to facilitate ongoing
analysis (Miles & Huberman, 1994). Transcription and initial open, descriptive coding were
usually done between sessions, along with such memoing as naturally arose (Creswell, 2007;
Miles & Huberman, 1994). Frequent memoing allowed the researcher to capture emerging
observations about what was happening while those thoughts were fresh. Coding was recorded
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using Atlas.ti data analysis software (Hewlett-Packard, 1993-2009). The unit of analysis was
one or more sentences or paragraphs focused on a single topic. As many codes were attached to a
unit as were deemed appropriate for an understanding of the important ideas present in the
situation.
As the study progressed, it became clear that some codes needed refining. For example,
at the beginning of the study, the a priori code ?student?s mathematics? was used. It arose from
the literature related to teaching experiments. When it became clear that many different forms of
mathematics were being observed, and it would be more profitable to code these occurrences
more specifically, that code was eliminated. Later on, the many codes which replaced it were
reduced to the particular mathematical ideas listed in the final coding guide.
After the interviews were concluded, initial coding for the sessions was completed,
including the addition of newer codes that had emerged during the course of the study.
Transcriptions were examined and further coded based on what was learned during the course of
the data collection, particularly with an eye to emergent ideas related to the subjects? interactions
with technology. Quotations were examined one by one to determine whether or not the coding
which had been done appeared to be sufficient.
Axial coding was used to categorize data into families, and selective coding was used to
connect cooccurring ideas and develop a view of what codes might be associated through
theoretical hypotheses (Creswell, 2007). This follows the pattern described by Glaser and Strauss
(1967) who noted that generation of conceptual categories is followed by hypotheses about the
relationships among the categories. Co-occurring codes lists were created using Atlas.ti and
examined to see which codes were associated with each other. Networks were created which
could show the density or connectedness of particular codes. Such a process served to help
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highlight emergent ideas. A one page code list was used at times to help with coding. The
software was also used at times to filter the codes according to family when examining a
quotation which seemed to contain information related to that family.
Queries done using the software helped create logical combinations of codes which were also
helpful in analyzing the data.
Indicative movements. I noticed during the course of the study that gestures and mouse
movements used to interact with the mathematical representations were more prevalent and
apparently more connected to the subject?s internal representations than I had anticipated. They
were also connected to emergent ideas related to the influence the technology was having on
student thinking. In order to go back and code for that emergent idea, I used the software?s
category searches feature, which allows the researcher to look for all instances of particular
words in the transcripts. Gestures were considered to be occasions where the subject pointed to
or indicated something by a physical movement in some way. A search was done through the
transcriptions for forms of the word ?indicate? or the word ?gesture.? Mouse movements were
considered to be occasions where the subject?s cursor movements could be tracked in some way.
Such movements were usually indicated in the transcription by the word cursor or mouse or
forms of the word trace or slide. Those words were used in a category search as well. Not every
single occurrence in the above named searches was coded as such; for example if it was a
movement of the interviewer and not the subject, it was not coded. In the final coding structure,
the instances of gestures and mouse movements were combined into the code indicative
movements which I felt more accurately described their role, as they seemed to be providing
indications of student thinking in similar ways.
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This review of the techniques used in the present study has included a discussion of case
selection, interview techniques, teaching experiment procedures and tasks, and data analysis
procedures. Attention will now be turned to the point of view I brought to the study as a
researcher and to the methods used to ensure that the research that was done was reliable and
valid.
My stance as researcher
It is helpful for readers to know any biases on the part of the researcher and to understand
clearly the role of the researcher in the study he or she is conducting. It clarifies exactly what
opinions of the researcher may have affected his or her objectivity or interpretation of data (Pratt,
2007). Readers of the present study need to know that I believe that it is essential that teachers
today understand and incorporate into their classrooms the power that technology has to help
students learn to love and understand mathematics. I have experience in the use of technology in
the classroom which supports that belief.
I taught entry-level college mathematics students for five years, many of whom had just
completed developmental mathematics courses. In this instruction I used technological
representations on many occasions, gaining practical experience with such technology and its
effect on student learning. I also used technology in teaching students in grades 8-10. At both the
college and public school level I provided websites through which students could gain access to
technological representations which would help them in their study of mathematics. I believe that
technology has the power to engage both the adults and the younger students I observed with
mathematics in a way that other techniques may not have.
In the public school setting in which I taught, computers were provided for every student
in every classroom. No training was provided, however, as to how the teachers of each individual
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subject would incorporate those computers into their classrooms. I believe that a greater
understanding of the importance of the use of technology in the mathematics classroom will help
school stakeholders to make wise decisions about what resources they provide for teachers.
It was helpful that I had worked with adult learners previously, as is desired in a teaching
experiment (Steffe & Thompson, 2000). I also had to observe what was happening and interact
with students in a manner that would allow me to learn how students operate and put aside my
own way of doing things as much as possible. I knew that there would be more happening than
teaching. Because I was interested in watching the development of students? thinking, observing
what I could of their internal representations, and watching the subjects? use of technology to
solve problems, I responded to their confusions or misunderstandings with particular types of
questioning rather than direct answers. I also at times allowed them to pursue mistaken ideas so
that I could see how the use of technology could allow those mistaken ideas to be revealed and
clarified. I made decisions as to what problems to present to the subjects so that emerging ideas
could be pursued, as opposed to making choices based on the completion of curricular goals. All
of these choices in the pursuit of emerging ideas about student thinking affected the course of the
research. In addition to clarifying any bias I had in the conduct of this study, additional measures
were taken to ensure its reliability and validity. A discussion of what was done and the
importance of this follows.
Reliability and Validity
It is vital that any study presented to the academic community be evaluated as to the
quality of the research which was done, in order for that research to be deemed worthy of
consideration. Such qualities may be considered under the broad headings of reliability and
validity. Reliability refers to the quality and clarity of the research techniques used. Validity
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includes the notions of internal validity or the authenticity of what it being said and external
validity or the potential of the research to be applied to appropriate settings (Miles & Huberman,
1994). Following is a discussion of constructs used to address reliability and validity in
qualitative research. After that the specific methods used in this study will be described.
Reliability and validity in qualitative research. Many constructs have been used to
address these issues in qualitative research and have been aligned with or added to the above
notions. Miles and Huberman (1994) provided a concise summary of many of those methods.
They drew together ideas from researchers including Guba and Lincoln (1981) and Schwandt
and Halpern (1988). Another researcher who drew from the work of Guba and Lincoln is Tuckett
(2005) who provided practical examples as to how the ideals proposed by Guba and Lincoln
might be enacted. He noted that Guba and Lincoln's (1989) trustworthiness criteria included the
notions of credibility, transferability, dependability, and confirmability and that their evaluation
criteria included credibility, fittingness, auditability, and confirmability. To help ensure
credibility, Tuckett (2005) used journaling, recordings, member checking, and triangulation,
among other things. To ensure transferability or fittingness, he included the use of thick
description, and purposeful sampling. To help ensure dependability or auditablity, Tucket (2005)
used field journaling, recordings, negative case analysis, and peer review. Field journaling also
helped him ensure confirmability.
Geertz (1994) described "thick description" as providing "meaningful structures" through
which people's actions may be better understood (p. 215). Thin description would merely
describe the action. Thick description would give more than just a description of the action. It
might provide, for example, information regarding the motivation for the action. Merely
observing someone whose situation seems to fit the idea you are searching for does not give you
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"the thing entire" (Geertz, 1994, p. 226). Part of the task is to "uncover the conceptual structures
that inform our subject's acts" (Geertz, 1994, p. 229). It is then hoped that this will generate
useful analysis.
Miles and Huberman (1994) described reliability as comparable to dependability or
auditability and said it was the quality which shows whether or not the process of the study was
?consistent, stable over time, and across researchers and methods? and possessed ?quality
control? (p. 278). In exploratory studies such as the one reported here, an analyst applies global
interpretations to the subject's mental processes, usually displaying transcript sections with his or
her interpretations. This assists the reader in seeing how analytical decisions were made
(Clement, 2000). Analysis must be conducted and reported in such a way that another researcher
can follow decisions made by the author (Chiovitti & Piran, 2003).
Rather than relying on multiple independent coders, the exploratory method relies on an
analyst who is sensitive to the subject and employs keen observation of detail (Clement, 2000).
If the researcher works alone, he or she may present in detail his or her perspective on the data in
such a way that readers can see things from the researcher's point of view, even if they do not
agree with it (Kvale, 1996). Kvale (1996) noted that readers should be able to "retrace and check
the steps of the analysis" (p. 209). Following is a description of measures taken to ensure the
reliability and validity of the current study.
Measures taken in the present study. During the course of the present study, the
following measures were taken which add to its reliability. A record was kept through memoing
and journaling of some of the questions and thoughts that arose during the course of the study
and its analysis. Multiple recordings were made of each session. Though only one researcher
was involved in this study, triangulation is present in the form of visual, verbal, and written data
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all being collected from the subjects, providing multiple mediums through which information
was conveyed. Peers provided feedback following the conclusion of the pilot study, a peer was
consulted during the course of the study for purposes of discussion and feedback as to the
progress of the study, and a peer was also consulted regarding the process of ensuring reliability.
Thick description and the inclusion of transcript portions are used in the description of
the cases. Following are additional measures taken. Included are a sample of researcher
questions, a look at the use of talk-aloud protocols, a description of how questioning was used to
ensure internal validity, and a description of measures taken to ensure external validity.
Questioning during the course of the analysis. One way analytical decisions may be
tracked is by knowing some of the questions which arose during the course of the research
(Chiovitti & Piran, 2003). Questions which arose during the course of this study included the
following examples, drawn from the researcher?s reflective journal and from memos recorded in
via Atlas.ti data analysis software (Hewlett-Packard, 1993-2009). Questions related to Marlon?s
case follow:
? Can technology help Marlon to see the connection between his idiosyncratic
representation and the standard representation?
? How can an understanding of functions be built on the idea of counting?
? What other mathematics does he possess?
? What connections is he making?
? Could [a particular use of variables] be considered an idiosyncratic
representation?
? What internal representations does he possess?
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? [H]e did not think of these two numbers as being added together previously.
Could this be related to his confusion about the coordinates of a point, which he
sometimes represented as an addition problem?
? What learning came from the technology and what came from pattern
examination? (Has he ever examined patterns like this before?)
Questions related to Marjorie?s case follow:
? How much did the software really help her to get this idea?
? How important was her own discovery and investigation with the software to her
understanding?
? How much information is there in what she says and does?
? What does this say about her internal representations?
These and other similar questions served to focus the researcher?s thinking during the
course of the study and afterwards as the analysis continued.
Inter-rater reliability. Unitizing was done semantically. Semantic units are chosen based
on the meaning of the text, while syntactic units are chosen by "graphic convention" (Murphy,
Ciszewska-Carr, & Manzanares, 2006, p. 3). Semantic units allow the researcher to encompass
whatever he or she feels constitutes a "complete idea" (Murphy et al., 2006, p. 4). Semantic units
allowed me to more clearly examine the meaning found in exchanges between interviewer and
subject. Since this study was designed to investigate student thinking, I felt it was vital that
meaning be a factor in the selection of units. Meetings were held with a knowledgeable peer to
discuss issues of reliability.
Murphyet al. (2006) noted the challenges of working with semantic units, which require
"interpretation and judgment on the part of coders" (p. 4). They said that "reliable and consistent
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interpretation and judgment between coders may not be possible in spite of training" (p. 4).
Creswell (2007) stated that in obtaining inter-coder reliability, it is important to decide what it is
that is to be agreed upon. It might be code names, code passages, or the selection of codes
assigned to the same passages. He noted that "there is flexibility in the process" (p. 210). In an
inter-coder agreement process he designed for data related to the HIPPA privacy act, he and
those working with him determined that they would not seek unitizing or coding passage
reliability. He said that, in the case of that particular research, to expect different coders to select
the same passages "would be hard to achieve because some people code short passages and
others longer passages" (p. 211). What they did do was look at passages that they had all coded
and see how well the codes they selected for those passages matched (Creswell, 2007).
I did conduct unitizing reliability tests with two different trained persons. I found that the
passages I selected were generally longer than the ones they selected, but there was some
consistency as to the location of the breaks. Of the places where I deemed that a break between
one unit and the next occurred, they also chose most of those same breaks (about 64%).
I also conducted inter-rater coding reliability tests, focusing on the four categories of
codes: mathematical content and thinking processes, representational ideas and issues, influences
and uses of technology, and other. I looked to see whether or not, given the same passage of
transcript, a trained peer with experience coding qualitative data would find those same themes
reflected in those passages. Using percentage of themes assigned by both coders that were
matched, the calibration session produced an agreement of 88.46%. A first independent session
produced an agreement of 77.78%. A second independent session by the same coder produced
an agreement of 78.95%. Total agreement over the three sessions was 81.94%.
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Internal validity. Creditability may also be described as trustworthiness or internal
validity and relates to the faithfulness of the description of the phenomenon (Chiovitti & Piran,
2003). It speaks to whether or not one should believe what the author says, (Schoenfeld, 2007).
It also indicates that the participants have in some way guided the process and that theory has
been checked against participants? meanings (Chiovitti & Piran, 2003). Creditable research
methods protect the researcher from inaccurately representing the subject?s intended meanings
(Chiovitti & Piran, 2003). In the present study, as noted above, a journal was kept. Some of these
thoughts recorded following the sessions provide a fresh memory of what occurred during the
session. The multiple video recordings also add to the believability of the study in addition to
providing evidence of reliability. This is particularly helpful as it allows key portions of the
resulting transcripts to be checked for accuracy. Triangulation, as noted above, also adds to
internal validity. Following is a closer look at two other measures of internal validity used in this
study, the use of talk-aloud protocols and careful questioning to elicit as accurately as possible
the student?s own thinking.
Talk-aloud protocol. Subjects were encouraged to talk out loud about what they were
doing as they worked, to talk continually as if they were thinking out loud. Instructions given to
subjects were adapted from ideas presented by Koichu and Harel (2007). At times the subjects
had to be reminded of the idea of a talk-aloud protocol, but each subject provided narrations of
their efforts during the course of their interactions with the technology. Such narrations, when
they were more than just a sentence or two, were coded as verbal streams and were common in
both Marlon?s and Marjorie?s cases. Campbell (2003) also referred to talk-aloud protocols and
noted that such techniques along with ?putting your mouse where your mind is? would help
researchers better capture their subject?s thinking (p. 74). Instructions were given to the subjects
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in the first session as to what they were expected to do. They were told to ?read the directions
and follow them . . . talk out loud about what you?re doing as you work, talking continually as if
you were thinking out loud.? Other interviewer statements occurring during the course of the
study to encourage this kind of talking were ?What do you notice? Talk about what you?re
noticing? and ?keep talking about what you?re thinking so that I?ll know how you?re thinking
about this.?
Questioning for validity. Questions were also used which were designed to support the
subject?s sharing of their own thoughts. Kvale (1996) noted that careful questioning during an
interview as to the meaning of what was said can help with validation. In this case, such
questions were noted by the following four sentences which were used as codes. Why? Explain
your meaning or choice. What do you see? What happened? Coding analysis shows that such
questions were also common in both Marlon?s and Marjorie?s cases.
External validity. External validity is also referred to as transferability or fittingness
(Miles & Huberman, 1994). Generality and importance ask how widely the research applies and
whether or not it matters (Schoenfeld, 2007). Generality does not imply importance. A study
may apply widely, but not contribute anything to our understanding of mathematics education
(Schoenfeld, 2007). External validity is aided by a clear description of the scope of the study, the
expectations for it, the subjects of the study, and by using thick description during the report of
the study. The recruitment and sample used for this study have been clearly described. It has also
been clarified that the purpose of this study is suggestive and not confirmative.
Any final decision regarding transferability rests with the reader (Chiovitti & Piran,
2003). The goal of theory development in mathematics education is to be able to communicate
ideas about it to the educated world in ways no other academic field can. In order to do so,
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theories must be developed and used wisely (Silver & Herbst, 2007). It is not within the scope of
this study to develop grand theory, but to formulate substantive theory related to the experiences
of these students which may suggest further work (Schwandt, 1997).
Another way to increase external validity is through the use of thick descriptive data.
This way judgments about fit can be more readily made by others (Lincoln & Guba, 2007). As
has been noted, thick description refers to the interlacing of meaning into described actions
which helps us to better understand those actions (Geertz, 1994). Case descriptions for this study
will include connections to ideas about representation and technology and other emerging ideas
which will help the reader to situate the subjects? actions theoretically.
Conclusion
In this chapter I have provided the reader with a description of the methods used for this
study, beginning with a look at the theoretical basis for the methods chosen. I discussed
qualitative research in general as well as qualitative research in mathematics education. I
provided information to support my choice of a teaching experiment and showed how such a
choice would allow me to learn more about the effect of the use of technology on student
thinking. Grounded theory was discussed as a foundation for data analysis. Following that
discussion, I presented a brief look at what I learned during the course of a pilot study. The
specific procedures used in this study were described, including the selection of subjects, the
instrumentation and data collection, and the technological procedures. I also discussed data
analysis procedures, and defined my stance as a researcher. In addition, I showed how specific
technological procedures in the form of a multiple camera technique allowed me to collect data
so that students? statements, paper inscriptions, and technological choices could all be examined
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in detail. Finally, measures taken to ensure reliability and validity were described. A description
of the results of the study follows.
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4. Results
The purpose of this study was to determine the effect of the use of mathematics
technology on adult developmental mathematics students? understanding of functions. Such
understanding was to be characterized by the quality of internal representations those students
appeared to be able to build as they interacted with technology. It was hoped that insight could
be found which would enable teachers of adult developmental mathematics students to help
those students overcome the challenges they may have, such as learning disabilities, a lack of
self-efficacy (that is, a lack of a belief in their own ability to be effective learners) or a lack of
understanding of what it is that is holding them back (Epper & Baker, 2009; Hall & Ponton,
2005; Wadsworth et al., 2007). It was also hoped that teachers of adult developmental
mathematics students might be provided with information which could broaden their use of
technology, which, as in other developmental situations, may not have been allowing the insight
into student thinking that would help these teachers better serve their students (Kinney &
Kinney, 2002). I conducted a qualitative case study in the form of a teaching experiment in the
hopes that it would allow me the best opportunity to examine adult students? interactions with
technology and study their thinking.
This chapter begins with an introduction to the subjects and some of their personal
characteristics. This will be followed by a summary of what happened during the course of the
teaching experiment. Following that summary the major theoretical ideas which arose will be
examined and supported by examples from the collected data on Marlon?s and Marjorie?s
experiences. The chapter will conclude with a summary of those ideas.
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Description of Subjects
Marlon, a 53 year old African American male, entered the study desiring to learn and
having persevered through personal challenges. He grew up in a large city in the northern United
States where he had experienced the influence of gangs before moving to a different school. He
dropped out of high school in his final year and then earned his GED while serving in the
military. The exact reasoning for his dropping out of high school was not clear. He said, ?I tried
to take my subjects very seriously but . . . trying [to] raise kids at that time I just fell out.? He
could not remember much about his high school mathematics classes, saying that he was
?probably more into sports? at that time. He did say at one point that he had loved mathematics
when he was growing up but that ?you really have to practice it all the time.? He now found
mathematics challenging, particularly after having been out of school for a while, which he said
made learning harder. He had not passed the first developmental mathematics course when he
took it during a recent spring semester, and stated that the heavy load from the English and
reading classes he was taking at the same time made it more difficult to find the time to get his
mathematics done. He took it again that summer and passed it. He has made some use of the
mathematics tutoring lab available on campus. He also expressed a strong desire to have his own
computer available at home. He had some experience with calculators and with software used by
his teacher which he said had been helpful.
Marjorie was a 36 year old African American female with a military background who had
left the military in order to pursue her education. Marjorie had apparently done well in school in
her childhood having been on the honor roll until the 10th grade. While Marjorie was in middle
school, her mother had become ill. For about 4 years the illness was not, apparently, life-
threatening. Towards the end of that time, after they had returned to the United States from
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overseas, her mother?s health deteriorated, and she passed away while Marjorie was in high
school. This personal tragedy probably contributed to Marjorie?s loss of interest in school for a
time. Even so, she graduated from high school with a B average. She said that ?I was able to
catch up with everything . . . and I graduated . . . with like a 3.2 but it wasn?t like . . . my 4.0?s I
was getting before.? After high school she joined the military. She ?didn?t want to do anything
with school? when she first got into the military, but about 3 years later she took some college
level courses. As part of her military service she was sent overseas where there was less access to
the courses she needed. She has continued to take classes, sometimes ?sporadically? and wants to
?get as much education as [she can].?
Table 10 provides a summary of the background information about the subjects of this
study. Both Marlon and Marjorie had military backgrounds and had experienced difficulties in
their childhood unrelated to their cognitive abilities which may have affected their academic
progress. Such challenges are in keeping with the non-cognitive factors that often hamper adult
developmental mathematics? students efforts in their return to school, such as personal demands
on their time (Gerlaugh et al., 2007). Marlon seemed to become more easily discouraged than
Marjorie, since he would often apologize for mistakes while Marjorie often asked to work on a
problem further. He demonstrated more of the lack of self-efficacy described in the literature as
being evident in some adult learners (Wadsworth et al., 2007). They both however, seemed to
have a strong desire to learn.
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Table 10
Subjects of Teaching Experiment
Marlon Marjorie
Demographics 53 year old African
American Male
Former military
36 year old African
American female
Former military
Secondary education Dropped out of high school
in his final year and earned
his GED while serving in
the military
Graduated from high school
with about a 3.2 average
Post-secondary education Had to take the first
developmental mathematics
course twice to pass it. Was
enrolled in the second of
three developmental
mathematics courses during
the study.
Had taken other college
level courses. Was enrolled
in the highest (third level)
developmental mathematics
course available during the
study.
Attitude Worked very hard to
analyze what he saw. Often
apologized for making
mistakes.
Keen desire to understand
everything. Wanted to work
to solve problems beyond
the designated time for the
session.
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Marlon Marjorie
Non-academic influences Experienced the influence
of gangs in his childhood
Lost her mother to cancer
while she was in high
school
The Teaching Experiment Sessions
Following is a summary of what occurred in the teaching experiment sessions. This
summary will focus on basic events and choices made during the course of the experiment and
not on the mathematical or theoretical results. Table 11, provided as a summary of the sessions
for both subjects, will be followed by a narrative description of the sessions.
Table 11
Content of Teaching Experiment Sessions
Marlon Marjorie
Session 1 He told about his background and
shared some mathematics he
remembered. He examined
?Looking at patterns? and ?Looking
at dot patterns?
She told about her background and
shared some mathematics she
remembered. She examined
?Looking at patterns? and ?Looking
at dot patterns?
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Marlon Marjorie
Session 2 He went back over his thinking for
?Looking at dot patterns?. He was
introduced to ?Another dot pattern?
and the software. He explored
graphing coordinate points with the
software.
She was introduced to the software,
and after free exploration she then
explored the xy plane, including the
graphing of coordinate points.
Session 3 He continued to explore the
graphing capabilities of the
software. He described the patterns
he saw in ?Another dot pattern?,
graphed the data points representing,
and made observations and
predictions about them.
She continued her exploration of
graphing points with the software.
She was reminded of her work with
?Looking at dot patterns? and she
continued that analysis.
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Marlon Marjorie
Session 4 After about 22 minutes of work to
remember what he had been doing,
he was asked to make predictions
about the number of dots in the 20th,
nth or xth
pattern. He was
introduced to the functions menu
and encouraged to try the functions
menu, using x as a variable and to
try creating a function which would
pass through the data points.
She explored the function menu,
and graphed some constant
functions. She was encouraged to
graph h(x) = x, put a measured
sliding point on it, and asked her
what it meant. She matched
algebraic and graphical
representations and looked for
intersections of h(x) with the
constant functions.
Session 5 With facilitation, he explored as he
tried to recall what we had done
previously. He was encouraged to
test ideas he was building about
functions.
She was challenged to graph a
function that would pass through
?Looking at dot patterns.? She
explored the effect of the change of
value of k in functions of the form
f(x) = kx. With facilitation, she
used the software to find the
equation of the function passing
through her data points.
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Marlon Marjorie
Session 6 With facilitation, he placed a vertical
line at x = 7 to help himself think
about what was happening on the
graph of the function g(x) = x + 9
with which he had chosen to work.
Once those two were graphed, he
was encouraged to examine their
intersection.
She was presented with a new table
of values, and asked to study it and
use the software to explore it.
Session 7 Using written instructions, he
created a dynamic representation
which included a movable measured
point attached to the graph of g(x)
that generated a table of values.
(Only 6 sessions held with
Marjorie)
Marlon?s sessions. The purpose of the initial interview session was two-fold. First I
asked him to talk about his background in a relaxed way. This provided me with information
about him and allowed him to become comfortable with the setting and situation. Second, I
asked him to analyze the patterns found in the two handouts ?Looking at patterns? and ?Looking
at dot patterns?, found in Appendix C. This allowed me to learn something about his
mathematical thinking and to provide content which the software would be used to investigate in
later sessions. Marlon attempted to remember some of the mathematics he was doing recently,
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but did not accurately remember how to use the devices of rote memorization he was familiar
with such as the first outside inside last (FOIL) method of multiplying two binomials, which he
misapplied. He looked at ?Looking at patterns? and ?Looking at dot patterns? during this session,
taking pains to examine them closely.
During the second session, I allowed him to take considerable time going over his
thinking for ?Looking at dot patterns? so that I could understand his thinking better. I then gave
him ?Another dot pattern? which was a simpler growing pattern, and he seemed to work with it
more fluidly. About 20 minutes into the second session, after his examination of ?Another dot
pattern? he was introduced to the software. He had not heard of Geometer?s Sketchpad. He was
allowed to explore the tools, and then he was introduced to the graph menu. I asked him to
choose graph and define coordinate system, asked him if he was familiar with it and what he
could tell me about it. I could then choose steps which let him become familiar with the software
and at the same time let me see him demonstrate and build understanding of graphing
representations. I had him use the point tool to place a point on the plane, use the measure menu
to find its coordinates, and use the selection arrow tool to move the point around so he could see
how the coordinates changed, predict what the coordinates would be in certain locations on the
plane, and test those predictions. This gave him practice selecting and moving objects and using
the menus. Later I had him use the plot points menu, which places a fixed point at one location,
so he could see the difference in the two types of graphing methods. I also introduced him to a
method for changing the scale of the graph.
In session 3 I continued to allow him to explore the graphing capabilities of the software
to try to recall what he had done the previous session for about 30 minutes. I then turned his
attention to ?Another dot pattern? before asking how we might use the software to explore the
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data found in the pattern. He described the patterns he saw. In his work on paper the data was
already being represented as a table of values and with some scaffolding questions on my part,
he graphed the data points with the step number as the value of x and the number of dots in that
step as the value of y. Once the points were graphed, he made observations about why they fell
where they did and what their relationship to each other was. He also made predictions about
where additional points would fall. By having him graph the points that were in the table, I
hoped to help him make the transition from a representation he seemed comfortable with (the
table) to a different representation (the graph).
During the fourth session, Marlon initially still had trouble remembering how to graph
with the technology. Helping him remember related mathematical vocabulary seemed to help
him find the right menu choices. He first plotted random points, even though I put the work he
had done on paper for ?Another dot pattern? in front of him. After about 22 minutes of
refamiliarization, we discussed how many dots would be in the 20th pattern and then in the nth
pattern or the xth pattern. Following this discussion, I introduced him to the functions menu of
the software. After some introductory explanation, I encouraged him to try creating some
functions which used x as a variable. Even though I knew his understanding of these
representations was weak, I wanted to see how much he could learn from using the software to
explore such representations. He created and graphed some linear functions, and made some
observations about them. I then challenged him to try creating a function that would pass through
the data points he had graphed. With some scaffolding questions designed to help him observe
what was happening in the things he was choosing to try, he was able to do so. I encouraged him
to write down what he had found and to take notes on the other ideas he had explored. After this
session I wondered how much he really understood about what he was seeing.
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For the next session, the fifth with Marlon, I started the session with a file already open
that showed coordinate points modeling data from ?Another dot pattern? graphed on the xy-
plane. The x-coordinate represented the step number, and the y-coordinate represented the
number of dots in that pattern. I also encouraged him to use the display menu to change the
colors of what he was creating so that the algebraic and graphical representations of the same
function would be the same ocolor. I let him explore a little before giving him his notes from the
previous session. He did not remember how we got the graph of the line that passed through the
data points and created some other functions as he tried to remember. I focused my questioning
on helping him think about what was happening and facilitating the explorations he was trying to
make to build understanding. With this facilitation, he was able to reason his way back to the
correct representation. I also encouraged him to test the ideas I saw him building about the
representations of functions by trying other functions which were similar to or different from the
ones he had been using.
In session 6, I wanted to challenge him to go beyond the understanding he seemed to
have about the representations associated with functions. About 30 minutes into the session, after
his explorations about the ideas he had been building, I facilitated his plotting of a vertical line at
x=7 to draw his attention to the fact that the variable x in the function could take on many
different values, such as 7. He had been thinking of x as zero. I hoped that looking at the
intersection of g(x) = x + 9 and x= 7 would help him see that the function g(x) crossed over other
places than the y-axis. I also hoped to give him entry into the meaning of the algebraic
representation by allowing him to find points on g(x) other than the intercepts on which he had
been focusing. In session 7, I gave him written instructions for creating a dynamic representation
which would include a movable measured point attached to the graph of g(x) that generated a
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table of values. I hoped to see whether this would help him make the next conceptual step from
noticing that the graph ?crossed over? particular places to seeing it as the set of all such
locations.
Marjorie?s sessions. As with Marlon the initial session with Marjorie allowed me to
learn something about Marjorie?s background. I also allowed her to share some mathematics that
she remembered and presented her with both the ?looking at patterns? and ?Looking at dot
patterns? handouts. I found that she saw the pattern present in ?Looking at dot patterns? more
easily than the other two subjects. Because she was the second subject in the study, it was several
weeks later before the second session with her was held.
For Marjorie?s second session I decided to start her with the introduction to the software,
rather than looking back at her previous work as I had done with Marlon. As with Marlon, after
some free exploration, I had her explore the xy plane in order that I might learn something about
her understanding of it even as she built understanding and continued to learn about the software.
The screen shot of the third session was lost due to a technical error. It started with
Marjorie looking at graphing points for a while until I felt she had done enough to be able to
move ahead and go to the next task. About 15 minutes into the session, I asked her to think back
to her work with ?Looking at dot patterns?. She had seen that it went up by 3 each time. I added
the idea of step number and asked her how else she might represent that data. After examining
the graph of the dots pattern data with the technology, I asked her to tell me how many dots
would be in the twentieth step. She made a general guess by looking at the technological graph
and then extended the representation in the table of values to test her guess. I asked her how
many dots would be in the 100th
pattern and she used the paper representation of the table of
values to solve this problem.
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In the fourth session, I had her explore the function menu. She graphed some constant
functions formed by entering a parameter in the formula. After she looked at the constant
functions, I encouraged her to graph h(x) = x, put a measured sliding point on it, and asked her
what it meant. I wanted her to see what understanding she had or could get from these
explorations before we went back to the dot pattern and table of values. After encouraging her to
match algebraic and graphical representations, I decided to let make her own choices as to what
she would do next. She wanted to find the intersection of h(x) with the two constant functions.
With minimal help she remembered what functionality and tools of the software would allow her
to do that.
At the beginning of session 5, I asked her to remember what she had been doing and had
two of her saved sketches ready to which she could refer. I challenged her to graph a function
that would pass through the graphed data points from ?Looking at dot patterns.? I reminded her
of the work she had done with the paper representations. Her choices led to an exploration of the
effect of a change in slope on the graph of a linear equation. After she had made connections
between what she found and the dot patterns, toward the end of the session, I assisted her in
placing a line through the graphed points and asking the software to find the equation of the
function for her and asked her to relate it to what she had already been doing. In session 6, I
presented Marjorie with a table of values, and asked her to study it and use the software to
explore it. Following these explorations, I asked her to give some concluding remarks about her
experiences.
Major Themes Arising from the Study
Data analysis began with ideas from the literature related to teaching experiments and
representation and a general search for places where technology use seemed to be important or to
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affect the student in some way. Through looking at the subjects? mathematical thinking, it
became apparent that particular misconceptions were repeated. It also became apparent that the
subjects had particular strengths in their mathematical thinking, such as the ability to observe
patterns and to reason and make sense of things. Identifying certain types of mathematical
thinking such as recursive thinking (noticing how patterns change from one to the next in a
sequence of patterns) was also important. The types of mathematical thinking that were observed
are collectively referred to by the category name mathematical thinking processes.
The importance of connecting multiple representations, the idea of a students? internal
representations, and the ideas of validity, usefulness, and endurance were codes arising from the
literature related to representation which were expected to occur. Mathematical language and
visual observations were codes addressing behaviors related to verbal and visual representations.
Idiosyncratic use of representations became apparent when the subjects used standard
representations in unexpected ways. The idea of indicative movements arose from observations
made about gestures used along paper representations and technological gestures made through
the use of mouse movements. Some studies were found which helped to underscore the
importance of these unexpected notions and assist in incorporating them into the present study
(Campbell, 2003; Stevens et al., 2008). The category name representational ideas and issues is
used to refer collectively to these constructs.
Some of the unexpected notions arising from an examination of representational issues
also gave rise to notions relating to what was being learned about the use of technology. For
example, examining indicative movements allowed the technology to become a window into the
students? mind, an idea Stevenset al. (2008) described in relation to their work with LOGO. It
also became apparent during the course of the study that technology was being used as an aid not
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only to reasoning, but also to mathematical communication, and to the use of standard
representations. Technology also allowed misconceptions to be revealed and cleared up. In
addition, the subjects were observed making choices which showed they were empowered by the
use of technology. These ideas have been collected under the category name influences and uses
of technology.
The reader may notice that there are additional codes listed in the coding guide. These
codes were helpful during analysis in understanding aspects of some of the other codes which are
used in the discussion of the results. They are not specifically addressed in this discussion,
because the ideas they represent are encompassed by other codes. For example, the ideas of
disequilibrium and equilibrium can be seen in the discussion of ways in which technology is
used to reveal and clear up misconceptions. The codes listed under the category ?other? were
used to identify data which did not illuminate the current theoretical investigation. They were
necessary because all data must be coded in some way.
Following is a more detailed look at the major ideas which arose from the study in each
of these three categories and how those themes were evidenced in the data. A look at
mathematical thinking processes will be followed by a look at representational ideas and issues.
Finally, the influences and uses of technology will be examined. Each section will include a table
which summarizes some of the data evidenced in the work of Marlon and Marjorie.
Mathematical thinking processes. As the study progressed, it became apparent that both
Marlon?s and Marjorie?s mathematical thinking processes contained misconceptions and also
showed evidence of their ability to reason. These ideas were important to understanding their
interactions with mathematics technology. Weaknesses seen in the subject?s mathematical
thinking processes included algebraic misconceptions, function and coordinate point confusion,
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and graphical confusions. Strengths included observing patterns and problem solving, reasoning,
and sense-making.
Algebraic misconceptions. Both Marlon and Marjorie showed evidence of algebraic
misconceptions that appeared to be hampering their mathematical progress. Marlon examined the
different representations carefully. At first he made connections and started to understand things
and then a misconception interfered with the understanding he was building. It became apparent
in the initial interview that Marlon had experienced some procedural instruction which did not
provide him with useful and enduring mathematical representations. He stated, for example, that
?I can always do this . . . this is the foil? referring to the FOIL (first, outside, inside, last) method
of multiplying two binomials, but then he applied this knowledge to the problem ????6464 ?? .
Figure 4 shows the representations he shared in the initial interview when asked to ?show? and
?tell . . . about? some mathematics of his choice that he remembered. His inability to properly
use a procedure with these algebraic representations is evidence of his algebraic misconceptions.
Figure 4. Representations Marlon created to show mathematics he remembered. Arcs indicate
that the FOIL method was applied to the multiplication problem.
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His lack of algebraic understanding could also be seen in his use of variables to explain
his mathematical ideas. Marlon observed that the sequence ?Another dot pattern? (Appendix C)
added one to the step number to get the number of dots. When I asked him how many dots would
be in the eighth step, he said, ?I would just, in this case . . . I would just add one . . . if it was 8
it?d be 9.? He also said in a later session that ?all I?m doing is actually adding a 1 to that and 10
it?s going to give me 11.? This was a correct description of the mathematics in the pattern. When
I asked him to express the same idea using n as the step number, however, rather than giving the
number of dots as n + 1, he reasoned that ?I?m just looking at n representing a certain number, a
pattern and o being the next letter in the alphabet. So it?s the same thing as far as the numerical
pattern. I would assume that the letter is going to be in a alphabetical order.? His reasoning was
logical and made sense in his mind, but showed that he lacked understanding of the use of
variables in mathematics. He could describe the pattern but he could not represent it
algebraically using a variable for the step number, which was further evidence of algebraic
misconceptions.
Though Marjorie had showed some ability to use algebraic representations in the initial
session, she also revealed weaknesses during the course of the study. She read f(x) = A as ?f
times x equals A.? Her use of the order of operations was weak, as she calculated 66 ??b by
saying ?6 plus 6 is 12 and you multiply that by B.? Her algebraic knowledge was also not strong
enough to allow her to go from insightful reasoning about the function which represented the dot
patterns to a corresponding algebraic representation. Although she noted that to get the number
of dots you multiplied the step number by 3 and added 1, she could not represent that idea
algebraically. As she considered such a possibility, she said,
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One times three is three but then you still gotta add and add one {she taps in the general
direction of the table of values and it is uncertain as to whether she meant to indicate a
certain value on the table} so if I was to try to do an algebraic expression, I guess I would
do it like that ? It?d be like
INTERVIEWER: Like what?
MARJORIE: One times x or um one x plus one equals y - something like that, because
I?m not quite sure exactly how I would write it as an algebraic expression
She did say ?the step? when asked what x would represent. Marjorie had the ability and
desire to think logically and solve problems, but there was a gap between her ability to think and
reason mathematically and her ability to use standard mathematical notation. This shows
evidence of misconceptions as to the purpose and function of algebraic representations.
Function and coordinate point confusion. A particularly important misconception that
Marlon brought to this study was his lack of understanding of the algebraic representation of a
function. Marlon consistently confused the representations of coordinate points and the
representations of functions. This was most commonly exhibited in his tendency to enter the
function f(x) = a + b in his attempt to graph a function which would pass through the point (a, b).
His confusion was also manifested in his representation of a coordinate point as a sum. When I
asked him to represent the point (0, 9) after its location was identified on a graph, he wrote 0+9.
When I asked him to write the same idea as if it were in a table, he said, ?In a table . . . it?d be
here, zero, plus nine? and entered 0 in the x column and +9 in the y column. Because of the
uniqueness of the confusion he exhibited and the persistence of the misconception, I have kept
this as a separate code from other forms of algebraic misconceptions, even though Marjorie did
not exhibit the same tendency. It also bridges algebraic misconceptions and graphical confusion.
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Graphical confusion. Marjorie exhibited considerable depth of observation and
deduction, but when confronted with standard graphical representations she had difficulty
working with them. As noted above, she did not seem to remember the idea of using a table of
values for graphing. In addition to that, during the introduction of the software, additional
weaknesses in Marjorie?s graphical preparation appeared. Note that some of the
misinterpretations may have come from misunderstandings about the terminology being used
and a lack of understanding of what was being asked of her.
She had constructed a circle and point on the circle. She had also used the software to
find the abscissa of the point on the circle, moved the point and noted that the value changed but
stayed negative ?because that?s where the circle is, on the negative side.? When I asked her to
predict what the y-coordinate would be, she at first hesitated then decided that she thought she
could. She said that ?when you?re doing graphing they always have those two, the x and the y: x
I think is always the starting point and y is the endpoint.? She seemed to envision the x and y
values to be starting and ending points of a journey. After finding using the ordinate choice on
the measure menu to find the y value of -8.22, she was unable to explain this number. She moved
her cursor toward (-8.22, 0) looked for other coordinate points near (8, 0). Later in the same
session, when she had a point with coordinates measured which she could drag around the xy
plane to observe how the coordinates changed, she had difficulty finding a location other than
(0,0) where both coordinates were the same. She was limiting herself in her search to the axes.
Even after experimenting and studying the movement of a point and the change in its
coordinates, she still described the x coordinate as ?the starting point? and the y-coordinate as
?where I want to get to . . . that?s my second point, my y.? When asked to explain what she
meant soon after this, she noted locations on the plane where points were positive and negative,
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observed changes in the displayed coordinates of points A and B that she was dragging around
the screen, but still could not really explain the meaning of what she was seeing. She compared it
to ?plotting coordinates on a map . . . . I know A is where you?re at right now, and then B is
where you want to get to.? She explained her concerns this way:
It is rather difficult to try to maybe explain what I do see . . . . I have an idea of why . . .
it?s negative, negative and it?s positive, positive, but it?s sort of a little difficult to try to
. . . explain . . . what I?m thinking right now because I really don?t know.
In session 4, she described the graphed points B = (-6, -6) and C = (18, 18) by saying ?it
gives me coordinates B and C and B is negative 6. {cursor to graphed point B which was at (-6,-
6)}And C is 18, positive 18 {cursor to point C which was at (18,18) and then over to the
coordinate point representations at the left of the screen}.? When asked to tell why two numbers
were listed for each point, she then indicated (0, 18) and (18, 0) with the cursor and said ?that?s
where it actually meets.? So at least by this time in the study, she seemed to understand what the
two coordinate values meant, but the language she used to describe the points was still
unconventional. Marlon also exhibited graphical confusion. For example, when asked to move
the cursor to a location where both coordinates of a coordinate point were positive, he moved the
cursor to the right along the x-axis.
Observing patterns, problem solving, reasoning, and sense-making. Both Marlon and
Marjorie were able to examine patterns, and make some sense of them. In doing so, they showed
their ability to reason and solve problems. Marlon?s ability to think logically was also apparent in
the initial interview, when he was asked to examine the pattern found in ?Looking at dot
patterns?. He examined it carefully and determined that since the odd patterns had the leg of the
T lined up with the center dot of the base but the even numbered pattern had the leg of the T
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lined up with the space between two dots in the base, that he would consider the odd and even
steps separately as if they were two different patterns. He was confused about whether or not to
count the center dot in the base of the odd patterns as part of the leg or not. His inconsistency in
doing so caused confusion as he analyzed the odd patterns. Focusing on just the even patterns
which he could more accurately analyze, he came up with a reasonable way of thinking about the
number of dots in the legs of the even numbered patterns. He gave the base of the T in his fourth
pattern 6 dots and the leg of the T in his fourth pattern 5 dots. He added 2 to the number of dots
in the leg of the second pattern to get the number of dots in the leg of the fourth pattern. This
reasoning does not fit the overall pattern as it would conventionally be analyzed, but it made
sense to Marlon based on his observations. Figure 5 shows Marlon?s work and to this has been
added some of Marlon?s statements about the even numbered steps as he saw them. Recall that
he was only given steps 1, 2, and 3 in the pattern to begin with and the rest he created. He had at
first ?estimated? that step 10 would have 13 dots in its leg. He later reasoned logically to
determine that he should remove the last two dots so that the leg would only have 11 dots.
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Figure 5. Marlon?s work in analyzing the pattern given in the handout ?Looking at dot patterns?
is supplemented here by a record of some of the statements he made as he was working.
Marlon?s focus on an aspect of the mathematics he could reason about amid a more
complex and confusing situation, and his selection of a unique pattern that he observed is helpful
background information in considering Marlon?s interactions with technology. It shows his
ability to observe patterns, solve problems, reason, and make sense of things.
When Marjorie examined ?Looking at dot patterns?, she first noticed the recursive
relationship, observing fairly quickly that the number of dots in each pattern increased by 3 with
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each step. She wrote down the expected number of dots for several more steps in the pattern and
drew a picture of the fourth pattern. With encouragement, she put a step number by each pattern
and also wrote the number of dots in each pattern. In the third session, after having been
introduced to the technology, she was asked to think again about the dot patterns and find the
number of dots in the 20th pattern. She said she?d have to ?count it out.? She could not think of
another way to represent the pattern that might be helpful. When asked if she remembered what a
table of values was, she said ?You mean (like) multiplication table(s)?? and when asked about a
table used for graphing, she did not seem to remember it. She was given a blank table and asked
to fill in the step number in the left column and the number of dots in the right column.
Once she had this representation to work with, she was asked again to find the number of
dots in the 20th pattern. She found this by filling in the table using the recursive idea of an
increase of 3 with each step which she had already noticed. She was later asked about the 100th
pattern and then she talked about being ?able to try to get there more quickly.? She studied the
table of values and noticed that there would be 61 dots in the 20th pattern. She said, ?You know
like 20 times three gives you 60, but that?s 61.? When asked to explain why she said that, she
explained ?I looked at the step {pointing to the 20 in the step number column} and then number
of dots, {pointing to the 61 in the #dots column}.? She was encouraged to write this idea down
and after she did so she said, ?It sort of works . . . because even with the ten . . . you can go . . .
times . . . 3, but again its 31, not 30. So it ends up so I guess maybe if I do add ? ooh . . .
multiplying the step by 3 then add one.? She looked at other entries and noticed that they were
also 3 times the step number plus one. As she was describing the relationship, she wrote it down
as ?multiply step by 3 then add 1 = # dots.? This episode shows the problem sovling, reasoning,
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and sense-making ability that Marjorie brought with her to the study and is important to consider
when examining her interactions with the technology.
Summary of mathematical thinking processes. Even though Marlon and Marjorie were
at different levels of developmental mathematics, there were commonalities in their
mathematical thinking processes. Both of them had valid mathematical ideas that they were able
to deduce from examining patterns in the data they saw. They were able to observe patterns, and
to solve problems, reason, and make some sense of what they saw. Both of them, however, had
difficulties expressing the mathematical relationships they saw algebraically. It is not known
whether or not Marlon or Marjorie possessed a learning disability, a condition which is not
uncommon to adult developmental mathematics students (Epper & Baker, 2009). It did seem to
be clear that algebraic misconceptions appeared to be part of what was holding back their
progress. Both of them also had difficulty at first in locating coordinate points, demonstrating
graphical confusion. Marlon?s mathematical thinking included unique a confusion about the
representations associated with functions and coordinate points which Marjorie did not share.
Table 12 provides a quick reference to some of the ways the constructs related to mathematical
thinking processes were evidenced in Marlon?s and Marjorie?s work.
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Table 12
Data Related to Mathematical Thinking Processes
Marlon Marjorie
Algebraic
Misconceptions
Misuse of algorithm
Inability to explain his valid
mathematical ideas using
algebraic notation
Misuse of the order of
operations
Challenges expressing her
valid mathematical ideas
algebraically
Function and
coordinate point
confusion
Lack of understanding of
algebraic representation of
function, confusing it with the
representation for a coordinate
point
She did not exhibit this
particular misconception
Graphical confusion Had difficulty when asked to
find a location where both
coordinates were positive,
moving along the x-axis to the
right.
Had difficulty when asked to
find particular types of
coordinate pairs, such as those
whose coordinates were both
the same
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Marlon Marjorie
Observing Patterns
Problem solving,
reasoning, and sense
making
Devised his own unique way
of understanding ?Looking at
dot patterns?
Deduced the functional
relationship in the table of
values for ?Looking at dot
patterns?
Representational ideas and issues. Representational ideas and issues were important to
understanding Marlon?s and Marjorie?s conceptions of mathematics. The ability of technology to
assist teachers in understanding their student?s conceptions would be an important addition to the
literature regarding the use of technology in teaching adults. A teacher?s current use of
technology may not provide enough depth of insight into student thinking (Kinney & Kinney,
2002). Important representational issues in this study which were used in an examination of
student thinking included the use of mathematical language, validity, usefulness, endurance,
indicative movements, multiple representations, and internal representations.
Mathematical language. Marlon?s use of mathematical vocabulary was sometimes
confused. On some occasions he used the word intercept to refer to the x and y coordinates of a
point. When describing the position of the point (7, -11), he said, ?I?m dropping down to a
negative 11 for my y intercept ? that?s where that point is located right there.? He also used the
word formula when referring to the representation needed to plot a coordinate point. After
entering 9 in the x column and -7 in the y column of a table, he said, ?then what (would) I have
to do is plot it (there) with the formula.? When I asked him to remember how he plotted points
with the software, he brought up the ?plot new function? menu rather than the ?plot points?
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menu which was in the same list. He also sometimes had difficulty understanding questions, for
example, when I asked him what the coordinates of an indicated point would be, he did not
answer correctly until he was asked what it would look like if written in table of values. Marjorie
also confused her mathematical vocabulary. She used the word coordinates on one occasion to
refer to the name and value of a parameter. On another occasion she used it to refer to a point on
the plane, saying that the software had given her ?coordinates B and C? referring to points B and
C. These episodes highlight the importance a student?s understanding of mathematical language
has to their use of technology.
Validity and usefulness. It is also helpful to note which representations the subjects
found to be valid or useful. Marlon, for example, seemed to find a table of values to be a useful
representation. It allowed him to consider the two elements of a coordinate point using a standard
representation rather than using a + b as representation of the point (a, b), as he sometimes did. It
also allowed him to observe multiple patterns in the function represented by ?Another dot
pattern.? When asked what he would put on the next line of the table of values for ?Another dot
pattern? after he had made some entries, he noted he would put 6 in the left column and 7 in the
right column, explaining that ?looking at the pattern here everything is in numerical order, and I
notice that the next one here {indicating the right hand column} follows 2 and it?s also starting
from 2 in numerical order.? It is uncertain whether the statement transcribed as ?follows 2?
meant ?follows the number 2? or ?follows also.? It does seem clear that he was noticing that the
right hand column started at 2 and that when looking down the column, the numbers were in
numerical order. Later he gestured from the left to the right hand column in explaining why the
eighth step would have 9 dots. These gestures, used to show where he was looking for his
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information, were followed by the statement, ?It?s just adding one.? The tabular representation
seemed to help him to see these relationships, and was therefore useful to him.
Marjorie?s work demonstrated how valid and useful representations developed for her
during the course of her work with the technology. At the end of session 5, after she had been
examining slope and seen that the function s(x) = 3x came very close to passing through the
coordinate points representing the dot patterns, Marjorie was shown how to construct a line
through the dots and use the measure menu to find an equation for the constructed line. She
noted that it was y = 3x + 1 and when asked how that related to what she had on paper, she said
?Exact same thing, because no matter what it?s a multiply of three and you?re always going to
add one to it.?
When confronted with a table representing a different, unknown dot pattern in the last
session she was able to describe its functional relationship fairly quickly. She noted that that
there was a ?difference of 2? and that ?you multiply that step by the difference of the dots and
add one.? She still did not know how to describe the relationship algebraically. She did use the
software, choosing from among ways that had been presented to her, and with some facilitation,
found the algebraic representation and then described how that representation related to the
functional relationship she had already described. This time she plotted the points for the pattern
using the point tool instead of the plot points menu and so the resulting equation did not exactly
correspond with the function describing the table of values. The software result was y = 2.02x +
1.04. She said ?that is the equation? and that ?it is letting me know . . . the difference between
the different points which there is a difference of two add one.? After I asked her what she would
write down if she were to write down what the equation should be, she said she would write ?y =
2x + 1?, pointing at and gesturing towards the computer screen as she did so. She explained:
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MARJORIE: Pretty much really exactly what um the equation is on there {pointing
directly at the computer screen} I'd probably have ah x equals or I guess like that y equals
2 x plus one {she paused slightly before the words "plus one" and gestured in the air
towards the computer screen as she said this}.
INTERVIEWER: Okay
MARJORIE: Because it?s a matter of {she pointed to the table of values on paper from
which the pattern arose} you're taking . . . how many steps you go {referring to paper}
and {gesturing over the table of values left to right across the table} if you?re going from
like number five, five steps, its 11 dots {she pointed at the 5 and then the 11} if you go to
six steps, its 13 dots, there?s that difference of two {note that the table stopped at 5, there
was a space and then step 20 was displayed - she gestured in the space below step 5} but
the way you get it would be five times, well . . . five times ten, I mean five times two is
ten add one is eleven. Six times two is 12 add one is 13. So it?s multiplying {she now
gestured in the air at the computer screen} two times the number add one.
She was making connections between representations. She saw the algebraic
representation presented by the software as confirming her understanding of the mathematics in
the functional relationship, and in that way it was valid and useful. It accurately reflected for her
the mathematics in the dot patterns and helped her communicate those mathematical ideas. She
understood that the coefficient of x in the technological representation told the difference
between the number of dots from one step number to the next and that x represented the step
number.
When I asked her to write on paper next to the table of values what the algebraic
representation would be, she wrote y = 5x + 1 and said ?but since I know what x is then?. When I
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asked her what x was she said ?x is two? and wrote y = 5(2) + 1 = 11. She was creating a
separate expression for each line in the table and using x as if it represented the slope instead of
the step number. For example, for step 5, she wrote y = 5x + 1, for step three she wrote y = 3x +
1 and explained that ?since you know that the difference is two, x is two.? When I asked her why
the software gave 2x + 1, she said ?Because the difference is two . . . the x represents um I think
the, the number or the step. 1, 2, 3, 4, 5, 6, .7. I think, I think that?s what the x represents here ?
the step . . . . The way I did it was the x represents the difference.? She knew that she and the
computer had used the variable differently and was able to explain those different uses. It is
uncertain why she used the variable on paper as she did, but the fact that she could clearly
explain the difference indicates that there was validity and usefulness in those representations for
her.
Endurance. Information about endurance can be seen when the subjects attempt to recall
what they learned in earlier sessions. When Marjorie was asked to consider the work she had
done in an earlier session, she was able to intelligently discuss the function she had created, f(x)
= (A -5) + 20, but could not remember entering the one I had asked her to enter, h(x) = x.
Regarding h(x) = x, she said ?I don?t remember where that came from . . . . I don?t remember if I
put that on there or not.?
When considering her work with f(x) however she said, ?I already had an equation
{cursor at f(x)=(A + 9) - 20} for um, f x and g x {moving cursor back and forth between those
two algebraic representations}. And I have A equaling 5 and I have B equaling 2.? Earlier she
had noted that entering 5 gave the function the value of -6 and that this number related to the
location of the graph in some way. It appeared that that she had a clearer memory of the
mathematical objects she created herself than the ones that I prescribed for her.
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She also remembered from one session to next that the scale of the graph could be
changed, although she could not always remember the technological procedures as to how this
was done. In session 4, when she was trying to clarify the location of a graph, she wanted to
change the scale. When I asked her what she was trying to do, she said, ?trying to get what you
had shown me about the moving it up and down . . . you can either make the [grid] squares larger
or make them smaller . . . so I could see more? The dynamic qualities of the technological
representation had endured for her in a manner that allowed her to call upon such an idea in order
to solve a problem.
Marlon also remembered dynamic representations, and from one session to the next could
remember how to change the scale of the graph. His verbal challenges, however, interfered with
his ability to remember some of the other technological procedures. When asked to remember
how to plot points, he said, ?Let me just, let me just go up to these here functions again just to
introduce myself again to them again {he moves the mouse along the top of the screen across the
menus and settles on the graph menu} I can go to the um I can plot a new function or plot new
function {he opens the new function window} just let play with it.? He continued to look at the
list of menus, at one point opening the calculate menu. I had to facilitate his recall of the correct
terminology in order for him to use the correct menus. When I told him the key word was
?point? he opened the plot points menu, noted that it looked familiar, and was able to continue
with his work.
Indicative movements and multiple representations. In response to my request that she
?show me some mathematics? in the initial interview, Marjorie settled on factoring the sum of
two cubes. In order to show this, she created her own prepared example by cubing 4 and 2 to get
the expression 64x
3
+ 8. She recalled the formulas enough to produce 4x + 2(16x
2
+ 8x + 4),
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which was not far from the actual solution of (4x + 2)(16x
2
? 8x + 4). As she thought about it she
at first wrote 4x + 2(x
2
+ 8x + 4). She then gestured with the pen in a bouncing fashion between
4x and x
2
and then over to 64x
3
. Eventually she inserted the 16, saying ?I knew I missed
something there.? The gestures may indicate that she was remembering that 4x should have been
multiplied by the first term in the trinomial to obtain the x
3
term in the original binomial. Other
events in the study showed how both gestural movements such as noted above, and movements
made with the mouse provided insight into the subject?s thinking beyond what their verbal
statements alone might suggest. Some of this influence of indicative movements has already
been noted in the discussion of the influences and uses of technology. Following are additional
examples which highlight the use of indicative movements to make connections between
multiple representations.
Marlon seemed to indicate that he was making connections between different
representations as he worked to represent the dot pattern found on paper using the standard
mathematical representations present in the technology. Figure 6 shows the situation at the time
this incident occurred.
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Figure 6: Marlon?s situation at the time he was connecting multiple representations on paper and
technology.
As he spoke about what he was seeing after predicting the location of (20, 21) and
plotting it using the technology, he looked back and forth between the paper and the screen. He
gestured with his pen along the paper representations and with his mouse along the technological
representations. The coincidental nature of these gestures may indicate that he was connecting
the different representations. A transcription of his dialog with actions is shown below.
MARLON: So again I?m looking at a numerical sequence {looking at the paper and
pointing with the pen from the step number 10 to the number of dots 11 written below it}.
And again if I assume that that top number is my x-axis {looking at the paper and
gesturing from the step number 10 written above the dot pattern to the space to the right
of it, then looking up at the screen} because it is running across the x-axis {cursor
moving along the x-axis} in a positive motion. My y is also in a positive motion being up
here {he looks down at the paper and then up at the screen during this phrase and moves
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the mouse to upper y-axis}, because these are also positive numbers going from the
center all the way to the top {mouse moves from near origin all the way up the y-axis}.
Note particularly here his phrase ?if I assume that that top number is my x-axis? and the
indicative movements and statements that followed. He appeared to be connecting the step
numbers used to label the dot patterns with the x-axis in the technological representation he was
seeing on the screen.
Marjorie also connected multiple representations through the use of technology. One
incidence of this occurred when she graphed f(x) = (A + 9) -20 and set A = 1. She noted that the
graph was located at y = -10 and by examining the algebraic representation was able to make the
connection that if A = 1, then (A + 9) - 20 = -10. In this way she connected the algebraic and
graphical representations. This incident will be considered in more detail as an example of the
way in which technology empowered Marjorie?s explorations.
Internal representations. One of the goals of the study was to determine the effect of
technological representations on the subject?s internal representations of mathematics. As the
study progressed, and it became clear that my time with Marlon was growing short, I decided to
give him a more dynamic representation in the last session which might build on what he had
been experiencing and effect his internal representations in some way. The activity presented
him with a prescribed set of technological instructions, since this would introduce a new feature
of the technology, and I wanted the technological steps to be clear. Even though the steps were
described, he still needed some help in interpreting those instructions, for example confusing the
selection arrow tool with the point tool (perhaps because it points to things).
The activity called for him to put a sliding point on the graph of f(x) = x + 9, create an
electronic table of values to track where the point was located, and eventually animate the point.
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Once the representation was created, he was free to study it and make observations. Marlon?s
language about what he was seeing seemed to change from the language he had previously been
using. During the study he had at first focused on the y-intercept of the graph. He then noticed
some other locations where the graph intercepted other parts of the xy plane. After seeing the
dynamic representation, he said ?when it was sliding up and down, it was actually giving me
these different locations {cursor to table and then graph} where it was crossing over the line.
Every single one was giving . . . in this case here x and y {cursor from line to table} . . . . So all
of these here are actually on this particular connected.? His mention of ?different locations where
it was crossing over? seemed to connect to his previous explorations. The rest of his meaning
was unclear. When I asked him what he was trying to describe, he said ?The whole line itself.? It
seemed that his understanding of the graph had moved to a new level. The sliding point and its
accompanying table seemed to have helped him to consider the idea of the entire line ? an idea
conceptually beyond the multiple points he had noticed previously. This is supported by his
statement that, ?The whole line comes from actually . . . connecting all the different points in a
straight line, connecting every last one . . . because they (were) plotted and they all . . .
intersected each other.? His internal representation seemed to have gone beyond a focus on the y-
intercept and ?intercepts? of other lines which the graph of f(x) crossed. He now seemed to
consider the entire line as a collection of connected points. The phrase ?every last one? in
particular seems to indicate that his thinking may have been broadened and gone beyond the idea
of ?crossing over.?
Insight into Marjorie?s internal representations of graphing was gained when she used
both verbal description and indicative movements to demonstrate how she found the location of a
coordinate point by movement outward from the origin.
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INTERVIEWER: So why does it list the numbers, why does it list two numbers for each
of those?
MARJORIE: Because its going off of, for C, its actually um 18 and 18 {indicating (0,18)
and (18.0)}and that like, that?s where it actually meets I mean if you were at the zero
{cursor at (0,0)} you would go to the right 18 {slides to the right along the x-axis} and
you would go up 18 {goes up to (18,18)} and that?s exactly where that point is.
Summary. Both Marlon and Marjorie had some difficulty with verbal mathematical
representations, confusing words on various occasions. This is in keeping with the literature
which tells us that adult learners? low literacy skills may hinder their ability to use technology
effectively (Li & Edmonds, 2005). They both were able to use tables to see patterns and solve
problems. Marjorie appeared to build validity and usefulness in her internal representation of a
function, eventually being able to describe in her own words how it modeled the mathematics of
the dot pattern it represented. Both Marlon and Marjorie remembered dynamic representations
which they wished to use. Both of them also had some difficulty in remembering technological
procedures, a representational issue, since recalling the verbal representation or mathematical
language associated with the technological representation was a key to recalling how to produce
it. Mathematical objects created by Marjorie endured better than those I prescribed for her.
Indicative movements were used by both subjects to connect multiple representations of the
same idea. Such movements also provided insight into their internal representations. These
qualities became important in examining the influences and uses of technology. Table 13
provides a quick reference to some of the ways that constructs related to representational ideas
and issues were evidenced in Marlon?s and Marjorie?s work.
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Table 13
Data Related to Representational Ideas and Issues
Marlon Marjorie
Verbal: Mathematical
language
Said ?intercept? to refer
to x and y coordinates
of a point, referred to a
set of coordinate points
as a formula, and
looked at the function
menu when trying to
plot points
Confused the use of the word
coordinates, using it on one occasion
to refer to the name and value of a
parameter and at another occasion to
refer to a point on the plane, e.g. ?It
gives me coordinates B and C?
referring to points B and C.
Validity and
usefulness
Found a table of values
to be a useful
representation
Built validity and usefulness in her
internal algebraic representations of
a function
Endurance Marlon remembered
how to change the scale
of the graph, but
misconceptions
interfered with the
endurance of other
technological
procedures
Mathematical objects she created
herself seemed to endure better than
mathematical objects I prescribed
for her to enter.
Dynamic qualities of helpful
technological representations
endured from one session to the
next, but the technological
procedures used did not.
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Marlon Marjorie
Indicative movements
and multiple
representations
Gestured along both
paper and technology,
connecting the step
numbers with the x-
axis values
Gestured toward related algebraic
representations as she worked on
paper.
Gestured to algebraic and graphical
representations as she explained why
the software placed the graph of f(x)
= (A + 9) -20 at y = -10 when A was
equal to 1.
Internal
representations
Expressed a clearer
vision of what a graph
was representing
following examination
of a dynamic
representation
connecting table and
graph
Demonstrated her understanding of
graphing coordinate points as a
movement from the origin. Also see
validity and usefulness.
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Influences and uses of technology. Marlon and Marjorie interacted with technology in
particular ways that showed how technology can be useful and influential for adult
developmental mathematics students in ways which add to its use in saving time, providing
individualized attention, increasing confidence, and decreasing anxiety (Kinney & Kinney, 2002;
Li & Edmonds, 2005; Taylor, 2008). Technology was used as an aid to mathematical
communication and as an aid to reasoning. It was also useful for revealing and clearing up
misconceptions. It provided a window into student thinking and aided them in the use of standard
representations. It also empowered both Marlon and Marjorie mathematically. These ideas show
how technology can be used in mathematics education and how it can influence student thinking.
Technology as an aid to mathematical communication. As he struggled with what was
presented to him in the initial interview, Marlon gestured toward the representations he found
and created on paper. Once introduced to the technological representations, he used the mouse to
point just as he had used his hands. The use of mouse movements provided a bridge for him as
he tried to communicate his thinking. An example of this occurred early in his struggles to find a
way to use standard representations to analyze the data from ?Another dot pattern.? During the
second session, I gave him ?Another dot pattern? to study and introduced him to the software. In
the third session, I asked him to graph data points from his table of values for ?Another dot
pattern? so the pattern could be analyzed with the software using standard representations. Figure
7 shows a screen shot of the graphed points.
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Figure 7. Marlon?s graphed points which arose from the dot pattern shown in the handout
?Another dot pattern?
Once the points were graphed he was asked if he noticed a relationship between them. He
said that the points were ?in a straight line? and ?going actually on a 45 degree angle.? When I
asked him why he said that, he said ?it?s just completely straight on a 45 degree angle? and then
he said, ?Well now it?s . . . I take that back.? He then used the mouse to explain why he was
changing his mind.
MARLON: Okay, here {tracing along the positive x-axis and then the positive y-axis}
right now I?m dealing with a - I would say a 90 degree angle.
INTERVIEWER: Okay
MARLON: 45 would actually have been here {tracing along the path where the line y = x
would be from the origin up to the right}. So it?s right off a 45 degree angle. So it?s not
completely a 45 degree angle . . . it?s a little bit off.
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He traced the x and y axis and noted that they were at a 90 degree angle and that if the
line going through the points had intersected the origin, then it would have been at a 45 degree
angle, but since such a line would intersect at (0, 1), he said it was ?just off of a 45 degree
angle.? In this way he was able to demonstrate with the mouse what his language was not
adequately expressing. He used the technology to make the genuine mathematical observation he
was making understood.
In Marjorie?s case, technology could aid her communication. In session 2, she was
observing how the movement of a point on the plane affected its coordinates. She moved the
point around the plane and described what was happening. At one point she described the point
A as being ?in a negative spot.? At the time, A was at (-2.57, 7.20). The technological
representation clarified her meaning, which otherwise would not have been clear. It thereby
aided her mathematical communication. In each of these cases, Marlon and Marjorie were using
the functionality of the software to accompany their description of mathematical ideas they were
trying to communicate. The ideas were clarified, and in this way, technology became an aid to
mathematical communication.
Technology as an aid to reasoning. Some of Marlon?s interactions with the technology
in session 4 seemed to indicate that he was beginning to reason logically based on the
representations he was seeing. He had graphed coordinate points for ?Another dot pattern? up to
(9, 10). When asked to explain what was happening and where the next point would be located,
he correctly located the point (10, 11) before graphing it and without directly referring to the x
and y-axis until asked how he knew where it would be. Soon after this, he described the pattern
he was seeing and gave the location of the next point, (11, 12), before graphing it. He described
what he understood about the pattern of points, accurately predicted the location of the next point
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in the pattern and graphed it using the technology, declaring, ?And there it is.? Later when I
asked him to predict where (20, 21) would go he went along the x-axis to 21 and then gestured
with the mouse in the general area where the point would be and then made a precise prediction
based on the locations of 20 and 21 on the x-axis and the y-axis. The use of technology allowed
him to make and follow up on mathematical predictions and so aided him in his reasoning about
mathematical patterns.
Marjorie?s work also showed the power of the technology to aid her in reasoning about
mathematical ideas. During the fourth session, Marjorie graphed the coordinate points
representing the sequence of dot patterns in ?Looking at dot patterns.? In the fifth session she
was asked to consider what function would result in the graph of a line which would pass
through those coordinate points. She noted that in an algebraic expression ?the letters represent
numbers? and that ?if we already know . . . what the numbers are that add up to the value . . . we
would just go and just replace . . . maybe one of the values.? So she decided to ?replace . . . one
of . . . the numbers in the equation.? We had recently been discussing the number of dots in the
50th pattern. As she considered what to replace with a variable, she said ?Probably the 3. Maybe
like have . . . 50 times x equals 151? Or 150?? After being encouraged to try it, she used the new
function menu to create q(x) = 50x. She wanted it to equal 150, but after some discussion,
graphed it and said ?It did something, but it didn?t . . . that don?t look right. That doesn?t look
right at all.? Once she saw what that function did, she tried changing the 50 to different values to
see what would happen. In this way the technology became an aid to her reasoning. She tried r(x)
= 70x, then s(x) = 30x. She then observed that ?the lower the number is, the more it moves away
from the y-axis.? She then tried t(x) = 10x. She said, ?Yes it does!? She moved the cursor back
and forth from algebraic to graphical representations as she explained what she had observed.
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Once she saw that a slope of 1 took her line beyond where she wanted it to be, she concluded
that ?I?ve got to . . . keep it between one and 10.? Eventually she tried 3 times x. She had
reasoned that the coefficient had to be between 1 and 5 and because of what she had seen in the
pattern, she tried 3. Because of the scale of the graph at the time, the graph of the function
appeared to land on the dots. As she discussed what had happened she mentioned that ?It was a
difference of three when [she] did the step and dots?, referring to the rate of change in the
pattern. The running conversation she kept up as part of the talk-aloud protocol helped provide
insight into her thinking and demonstrate that she was engaging in reasoning and problem
solving with the software. She was noticeably excited when it behaved the way she expected it to
behave, and technology was an aid to her as she reasoned her way toward her conclusions.
Using technology to reveal and clear up misconceptions. Both Marlon and Marjorie
experienced episodes where they discovered misconceptions through the use of the software and
ideas were clarified. Two different episodes are used below to illustrate how this happened in
Marlon?s case. An episode from Marjorie?s case follows.
Marlon?s case. Because he confused the representations of functions and coordinate
points, he used f(x) = a + b to try to graph a function which went through the point (a, b). During
the fifth session, I asked him to recall the function he had previously found which, when
graphed, produced a line which passed through his data points. He could not remember what that
function was, and in his efforts to remember, his confusion over the representations for
coordinate points and functions interfered. He opened the function menu. He knew that the
function had to pass through the point (1, 2) and so he had graphed f(x) = 1 + 2. Since the graph
did not travel in a diagonal line through the points, he tried again and looked at the currently
graphed point which was farthest to the top and right of the graph at that time, (8,9), and graphed
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g(x) = 8 + 9. This graph of g(x) turned out to be out of the viewing window, and he had to
change the scale of graph to see where it was. He did this himself with no facilitation after he had
done some additional exploration which had produced h(x) = 8 + 9 and the equation 1 +2 = 3.
When he finally saw the graph of g(x) he said ?I?m getting a straight edge again here? and
indicated 1+2 = 3 and then the graph of g(x) = 8 + 9 with his mouse. When I asked him where
the graph of g(x) had come from, he at first replied ?this last one I just put in? and indicated 1 +
2 = 3, then said ?as you were? which was a phrase he commonly used when he realized
something was wrong. He then counted to see that g(x) crossed the y-axis at 17. I asked him
?Where might 17 come from?? This was genuinely puzzling to him, and he wondered aloud
?How did I get 17 in there?? I asked him if there was anything on the screen that might give him
17. The situation at the time he responded is illustrated through the image presented in Figure 8,
which is followed by his statements.
Figure 8: These portions of a screen shot, from which non-essential elements have been
removed, show the situation at the time Marlon was asked to consider why the graph of h(x) = 8
+ 9 might have been placed at y = 17.
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MARLON: {he put the cursor at h(x) = 8+9 then moved it down toward the bottom of the
list between q(x)=x+9 and 1+2=3} ,ummmm, {He moved the cursor back up the list and
then to 1 + 2 = 3} I don?t see, I mean, that would give me 17. How?d I get that one there?
{cursor near 1+2=3 and q(x) = x + 9} And this is the, this is the same -similar to the one
that I gave down here {cursor at (0,3)} through 3. Okay {cursor to 1 +2 =3} [18:48]
{cursor up to h(x) = 8 + 9}. Oh, not unless these added together.
By looking at the multiple representations presented by the technology, he realized that in
the functional notation f(x) = a + b, those two numbers were in fact added together to determine
where the graph would be located, and were not representative of the two numbers describing a
coordinate point, which was the way he had been trying to use them to graph the function. The
location of the graph of h(x) = 8 + 9 in a different place than he expected it to be revealed the
misconception and examining the different representations present helped clear up the confusion.
Even though the representation 1 + 2 = 3 was not the functional representation which matched
the graph which passed through (0, 3), the presence of that representation may have been
important to his building an understanding that ?these added together.?
In session 6, I sought to challenge Marlon?s understanding of functions of the form f(x)
= x + b. He seemed to have an interpretation of them which was restricted to the location of the
y-intercept. I had also learned in the pilot study that the use of a vertical line at a particular x
value could help developmental mathematics students to visualize more clearly relationships in
the graphs they are seeing. I facilitated Marlon?s construction of the line x = 7 and asked him to
notice where the graph of the function g(x) = x + 9 intersected that line. He found that it was (7,
16) by examining the graph and looking to see which x and y axis locations would give him a
point at that intersection. He did not seem to understand the algebraic representation. When
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asked where the 16 came from, he said that it came from the y-axis ?where it intercepts . . . here
again on the g of x?, indicating the point (7, 16) and the algebraic representation of the function
and the point again, but he did not say anything which indicated that he connected these
representations with the idea of adding 9 to 7 to get 16. Earlier, I had asked him to think about
some other points that would land on the graph and asked him to think about the algebra and
consider what it represented. He said, ?x is actually zero {cursor to (0,0)}
. . . and plus 9 is {cursor to algebra and back to (0,0) then up y-axis to (0,9)} plus 9 on my y a -
intercept.?
When I asked him what else we knew about the line, he pointed out the x-intercept. When
asked to graph a point that would land there he tried (0, -9), decided he hadn?t done something
right and opened the plot points menu again. He began again by saying x was going to be zero
and I asked him why. As he explained his thinking and moved the cursor to show that thinking,
he was able to realize his own error.
INTERVIEWER: How do you know x is going to be zero?
MARLON: Well if - you know its, this is my x intercept again {cursor up and down the
x-axis} so I know it?s right there {indicating (0,0)}. And then if I go to a negative 9 . . .
Oh {cursor down to (0, -9)}. Okay, um, {cursor back at (0,0}. If it?s going to be x . . . is
it going to be . . . {looking back at paper representation, pointing with pen to (0,-9)} I?m
looking at this is going to be . . . {puts pen down and looks back at screen}. I?m trying to
? I?m thinking that it?s actually, in order for me to get it here {cursor at (-9,0)} (I?m on)
a negative 9 on the x-axis. So if I give it x being zero its going to start here {cursor at
(0,0)} and (then) say negative 9 it would have brought me down here {cursor at
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(0, -9)}. So I?m thinking now I got to reverse it in order to get it over here. {cursor at (-9,
0)}.
Note that other than (0, -9), none of the other points he had written on his paper had an x-
coordinate of zero. Because the technology graphed the point (0,-9) in its correct location,
Marlon was able to recognize that his idea that it was the x-intercept was a misconception. He
examined the representations further and was able to find the correct coordinate point value for
the x-intercept, clearing up his misconception.
Marjorie?s case. Marjorie?s use of technology also allowed some of her misconceptions
to be revealed and helped clear some of them up for her. This can be seen in her work in
graphing coordinate points. I asked her to fill in a table of values on paper representing the
sequence of dot patterns with the step number in the left column and the number of dots in the
right column. She was then asked to graph those points. She dragged a point using the point tool
to graph the point (1,4) and placed it at (4,1). I told her to use the ?plot points? tool instead so
that the points would stay where we wanted them to be. When she did so, the software placed the
point in its correct location, and she was able to see her mistake. She was also able to give a clear
description of what that mistake was. She said, ?I went to the right hand side and . . . I just
moved up from the center - I moved up one, and to the right, to the right four. But in actuality . . .
I should have moved to the right first and then up four. So . . . it was right and up.? In this way,
the technology helped reveal and clarify a misconception.
Technology as a window into student thinking. An examination of Marjorie?s graphing
work also shows how technology can provide a window into student thinking. This adds to the
idea of technology as an aid to mathematical communication by showing how technology can
reveal internal representations and ways of thinking about mathematics that might not be
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apparent from the student?s verbal descriptions alone. Stevenset al. (2008) in their work with
LOGO, noted that examining student?s problem solving processes can provide a ?window in to
the student?s mind? (p. 199). Here, the window into student thinking provides a view of their
internal representations of mathematics.
For example, when I asked Marjorie to find a point whose coordinates were the same, she
moved the cursor from its current location at about (6, 9) over the y-axis and down to the origin.
Later during the same session, when I asked her to find another such point she said, ?Both the
same, let?s see. Hm. Actually I cannot. Not where they?re both the same. Right there at the
origin.? Here is the same quotation with mouse movements inserted.
Both the same, let?s see {moves the cursor up to (0, 13), which was the maximum y-axis
coordinate}. Hm. {down to (0, 0) and over to (-10,0) and to (22,0) which was the
maximum x-axis coordinate, back to (0,0) and down to (0,-13) which was the minimum
y-axis coordinate}. Actually, I cannot {cursor back to (0,13)}. Not where they?re both the
same. Right there at the origin. {cursor is now back at the origin}
The indicative movements described reveal that she was restricting her search to the axes,
something that would not have been apparent from her words alone. In this way, the technology
has provided a window into her thinking. The episode noted earlier in which Marlon is
explaining what he means by ?just off of a 45 degree angle? is also an example of how
technology provides a window into his thinking. The indicative movements show what he meant
and how he thought about the relationship of different angles beyond what his words alone
would have told us. He chose aspects of technology to help him communicate, and this
communication provided a window into how he thought about angles on the xy-plane.
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Technology as an aid in the use of standard representations. After she failed to find a
place other than the origin where both coordinates were the same, I asked Marjorie to find a
point for which the right coordinate was bigger than the left. She said she could do that ?By
going into the negative? and moved the cursor to (-11, 0). After being asked ?And what happens
if you go up?? she went first to (-13, 9), observing that the right coordinate was increasing as she
went up. She then began to explore more freely and moved the cursor to (-12, 12) without
prompting, noting that ?I?m at 12, 12. It?s the exact same . . . .? After being prompted that this
was true except for one thing, she said:
MARJORIE: The negative and if I probably did it for the opposite side it?d be the exact
same thing except for . . . the left it would be positive 12 and the right would be negative
12.
INTERVIEWER: Why don?t you try . . . {she moved the cursor to (12, 12)}
MARJORIE: Yep
INTERVIEWER: Now is the . . .
MARJORIE: Ooooh, no! They?re bo ? okay, okay. They?re both positive because I?m in
the positive area {pointing to the screen}. If I go down it?s negative {cursor at (12, -8)}.
These appeared to be genuine discoveries for her, even if as reminders of knowledge she
possessed in the past. Note that once prompted to move off of the axes, she then made additional
choices about what to explore. In addition to the interaction noted above, she made general
observations about the xy plane, moving a point around the plane to confirm her observations,
noting for example that ?when I drag the A over to the left hand corner, the first number is
negative.? The technology had been an aid to her in the use of standard graphing representations.
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Technology also helped Marlon in his use of standard representations. For example, when
Marlon saw that h(x) = 8 + 9 passed through the y-axis at (0,17), he was able to make a
connection between the algebraic and graphical representations. Later, he was able to create a
function on his own using a similar representation.
Technology was also an aid in the use of standard representations in Marlon?s work on an
occasion when he was trying to find a function which would pass through the data points he had
graphed for ?Another dot pattern.? He had tried r(x) = x + 2, and I asked him to think about what
he had done.
MARLON: Okay so now what I did especially is I provided x plus 2, that?s what gave
me this {he traced that graph}.
INTERVIEWER: Okay
MARLON: And again, x , this is my x axle {indicating the x-axis} I mean axis and plus
two gave me right here {indicated (0, 2)} [23:46]. Okay. So if I want to do this here I can
actually say x + 1 {indicated (0, 1)} so if I go to . . . the graph a new function {he opened
that menu} I can say x plus 1 {he entered it}.
He had not chosen plot new function. Once the function was plotted he said.
MARLON: There she go. x + 1 and the reason why I got it is because here again is my x-
axis {traced the x-axis} and plus 1 is right here {he went up to (0,1) from (0,0)} and it
intersected through those points that I graphed last week {he traced along the graph}.
Note that he used the cursor to indicate the location of the previous attempt and that once
he graphed the new attempt he used the cursor to indicate and trace along key aspects of the new
attempt. It may be that such movements serve to help the learners solidify in their minds the
knowledge they are building.
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Other instances are indicative of a focusing of the subject?s mind as he or she examines
representations. This could be seen when Marjorie changed the parameter A to equal 5 and then
looked to see its effect. She moved her cursor along the algebraic representation containing A as
she spoke aloud, saying ?Okay, let me see . . . . 5 {cursor at algebraic representation, and she
moved her cursor over it as she spoke} 14, and then I have to, because I did my parentheses first,
it was 14, and 14 minus 20 would make that negative 6 because 14 minus 20 {looked up in the
air in thought}, negative 6.? She did look up in the air in thought at one point here, but she also
moved her cursor along the representation in the process of trying to understand it. She then said,
?And negative 6 is on the graph {cursor back and forth at the graph} negative 6.? Here she was
back at the technological representation making connections and building understanding of
standard representations.
Empowerment through the use of technology. Marlon was making connections between
different standard representations. He used the technology for his own explorations of ideas he
was having about what was happening and worked to understand the functions he was creating.
In this way technology empowered him mathematically. Two occurrences in session 5
demonstrate this.
Examples of empowerment from Marlon?s work. Marlon?s response to being asked to
graph a function parallel to those he had seen (that is, a constant function) is a good example of
the way he was using the technology to build on his own unique understanding. He decided that
he wanted a function to go horizontally through the point (0, 14). Rather than entering f(x) = 14,
even though he had seen that the numbers a + b in his other examples were ?added together?, he
entered f(x) = 7 + 7. He said, ?I would actually go, go to plot new function {which he did} and
lets say I want to do 14. I would just say 7 + 7 again {he entered that expression into the function
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and hit okay} and it should give me straight across.? He couldn?t quite bring himself to abandon
his internal representation of f(x) = a + b, but yet he understood something more than he had at
the beginning. He decided himself that the horizontal line he wanted to graph would be at y = 14.
He decided himself to use 7 + 7 (rather than something else for example such as 8 + 6), and he
was able to see that what he thought would happen did in fact happen.
During session 4, I asked him to enter the variable x into the functions he was graphing.
He graphed f(x) = x + 9 and g(x) = x - 9 and noticed that they crossed the y-axis at (0, 9) and (0,
-9) respectively. He was building some understanding that functions of the form f(x) = x + b
cross the y-axis at (0, b). He also observed that the two graphs were parallel. When I questioned
him further, he was able to deduce and demonstrate that f(x) = x + 1 went through his graphed
points. In session 5, he had reasoned his way back to that representation again after forgetting it.
Later in session 5, after some discussion, I asked him to try something that he hadn?t
done before. He entered v(x) = x ? 9 and then said ?Could I add more?? and I told him he could
try that. The software automatically entered parentheses to what he entered to give the function
v(x) = (x ? 9) + 6. After giving the graph and equation matching colors, he moved the equation
close to the graph, studied it and said, ?Now how did I get that one?? I turned the question back
over to him. During the exchange he moved the cursor from the algebraic representation to just
below the y-intercept briefly, back to the algebra and then to a blank spot in the second quadrant.
INTERVIEWER: Think about what?s happening with that one. Why is it going the way
its going?
MARLON: Okay. x negative 9 - so - x negative 9 here - x negative 9 {cursor at algebra}
? I?m coming across here to a 3, {cursor at (0,-3)} Ah! What?s happening {cursor at
algebra until the word ?giving? when it moves to (0, -3)} is that it?s subtracting {cursor
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moves back and forth along the algebra until ?negative 3? when he moves it back to (0, -
3)} the negative 9 from the 6 and it?s giving me a negative 3 and that?s the reason why
it?s intersecting here {indicates (0,-3)} because . . . its subtracting the -9 from a positive 6
which gives me actually negative 3 and again it is diagonal.
His cry of ?Ah!? seemed to indicate confidence as did his clear explanation. Although
there was most likely much he still did not understand, he understood a correct mathematical
idea which made sense to him and was built on his own demonstrated prior knowledge. He had
chosen the exploration himself, he had put his plan into action using the technology, and he had
drawn a correct conclusion about the connection between the algebraic and graphical
representations. Such an exploration may have been difficult for him to do without the use of
technology, and in this way he was empowered by his use of technology.
Examples of empowerment from Marjorie?s work. Technology also empowered Marjorie
to explore her own mathematical ideas. This can be seen in her explorations of the function
menu. One of the first things she did in this exploration was to create the parameter A = 1. She
then used that in the function f(x) = (A + 9) ? 20 and graphed that function. She noted that if one
were to substitute in the value of 1 for A, f(x) would be -10. At first she didn?t seem to notice the
graph of the line. When I pointed it out to her she noticed how far to the left and right the line
went. I asked her to consider where it was located ?up and down? and she said ?looks like
negative 9? and wanted to change the scale of the graph, but couldn?t remember how to do that
as she had been previously instructed. She said that she was ?trying to get what you had shown
me . . . you can either make the squares larger or make them smaller.? By ?squares? she was
referring the size of the grid spaces on the coordinate plane. In this instance, she knew what she
wanted to do with the technology. She wanted to use a feature she had previously seen, changing
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the scale, in a way that would help her to understand the mathematics better, but she needed
some facilitation. Once she was able to change the scale of the graph, she was then able to
observe that ?it?s at negative ten.? When I asked her why the graph was located there, she moved
her cursor to the algebraic representation. Notice her statement and mouse movements as she
thought about how the algebraic and graphical representations were related to each other.
MARJORIE: Ah, you know I really don?t know. Because up here, {cursor running back
and forth across f(x) = (A + 9) - 20} well, wait a minute, ten minus, I guess it would be
negative ten. Yes it would be. {cursor near the A in that function}. Um, the equation for
the function of x is negative 10 once the equation?s worked out. It is negative ten, {she
has been moving the cursor along the algebraic representation of f(x) as she speaks and
now moves it to (0, -10)} so that?s where it plotted . . . at negative ten. Let?s see {goes to
graph menu}. Graph, plot new function {cursor to that choice, but doesn't open it}.
At first she said she didn?t know, then she said ?wait a minute? and made a logical
deduction. She appeared to consider and connect the different representations and then headed to
the menu with which she could create another representation to check her understanding.
Though invited to record on paper what she was experiencing, she instead stayed with the
technology and created 66 ??b , with B = 1. Even though she had some difficulty with order of
operations, the fact that B was 1 in this instance concealed that weakness, preventing it from
being revealed. The graph was at y = 12, where she expected it to be. She was making
connections between different representations using her own mathematical choices which were
empowered by the technology.
Later in the study, she changed the value of B and was able to see that something was
wrong in her understanding. With B = 2, she expected the graph to be at y = 24, but it was at y =
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18. After being asked to consider how that might happen, she noted that ?you replace the B
which equals 2, you?re gonna put the 2 in there.? After being asked what she would do then, she
said, ?I guess you could, two times 6 which is 12, and then add 6 to 12 and we get 18. Hm.
Okay.? Once she realized this, she again chose to explore further. She changed the parameter A
to 5 so that she had f(x) = (A + 9) ? 20 with A = 5. She said
MARJORIE: Let?s change that parameter for A . . . let?s see, let?s change that to 5. {after
changing it to 5, she hovered the cursor over the ?ok? button for 5 seconds, then clicked}.
Okay, let me see if I (know), 5 {cursor was now at the algebraic representation and she
moved her cursor over it as she spoke} 14, and then I have to, because I did my
parentheses first, it was 14, and 14 minus 20 would make that negative 6 because 14
minus 20 {she looked up in the air in thought}, negative 6
INTERVIEWER: Okay
MARJORIE: And negative 6 is on the graph {cursor back and forth at graph} negative 6
Even though she was exploring and building some understanding, there were still
limitations to what she knew about the representations, as revealed by her statement that ?B is at
12, positive 12 and A is at negative ten because it?s going off the equations.? Rather than naming
the functions f(x) and g(x), she was naming them by using the parameter which was included in
their argument. She did, however, move her mouse between f(x) and g(x) and describe the
parameters accurately. During the following session, when looking at her saved work from the
session in which she had created f(x) = (A ? 5) + 20 again, she was able to discuss with some
understanding the function she had created, namely f(x) = (A ? 5) + 20, but could not remember
the function I had told her to enter, namely h(x) = x, which was in the same sketch. Her ability to
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chose an exploration, learn from that exploration, and then in a later session intelligently discuss
her own creation is evidence of the empowerment technology is capable of providing.
Summary. Both Marlon and Marjorie used technology in their explanation and to reason
about mathematical ideas. As they did so, misconceptions were revealed and cleared up, and they
both learned more about standard representations. Their work provided a window into their
thinking. They were also empowered in that they followed their own ideas, creating functions of
their own choice for their investigations. Even though their work was at different levels, this
empowerment could still be seen. Table 14 provides a summary as to some of the ways the
influences and uses of technology were evidenced in Marlon?s and Marjorie?s work.
Table 14
Data Related to the Influences and Uses of Technology
Marlon Marjorie
As an aid to
mathematical
communication
Used cursor movements in his
explanation of why he said the
line through the points would be
?just off of a 45 degree angle?
Used technology to explain the
positive and negative aspects of
the xy-plane allowing us to see
what she meant when she said a
point is ?in a negative spot?
As an aid to
reasoning
Deduced the location of
additional coordinate points for
?Another dot pattern?
Explored the influence of k in
functions of the form f(x) = kx
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Marlon Marjorie
To reveal and
clear up
misconceptions
Gained a better understanding of
functions of the form f(x) = a +
b. He also discovered the correct
coordinates for (-9, 0), which he
has mislabeled as (0, -9)
Discovered that she had
misplaced the point (1,4) at (4,1)
and found its correct location
As a window
into student
thinking
When communicating his idea
that the graphed points were
?just off of a 45 degree angle?
his cursor movements allow us
to see his thinking
Cursor movements showed she
was only looking along the axes
for a point of the form (a, a)
As an aid in the
use of standard
representations
Saw that h(x) = 8 + 9 passed
through the y-axis at (0,17) and
was able to make the connection
between the two representations.
Found a point of the form (a,a)
other than the origin and observed
how the coordinates changes as
she moved the point around the
xy plane
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Marlon Marjorie
Empowerment
through the use
of technology
Created the function f(x) = 7 + 7
to pass through (0,14)
demonstrate his growing
understanding.
Created a function of the form
f(x) = (x ? a) + b and explained
that it passed through the y-axis
at ?a+b
Created constant functions using
parameters and explained how the
value of the parameter affected
the location of the graph
Chapter Summary
Three major categories of information emerged from this study: the importance of
mathematical content and thinking processes in the use of technology, associated
representational ideas and issues, and particular influences and uses of technology as related to
student thinking. This chapter has examined some of the ways those themes emerged from the
cases of Marlon and Marjorie. Even though their mathematical understandings differed in some
ways coming into the study, they were able to use the technology in closely associated ways to
build greater understanding of representations associated with functions. The themes which
emerged in the study appeared to converge in many ways. Indicative movements provided
insight into their internal representations. This was evidenced in the ways that Marlon and
Marjorie appeared to think about what was happening, and what they appeared to understand. By
considering the mathematical thinking and representational ideas present in the technological
interactions, the issues and uses related to the use of technology could then be clarified.
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Technology?s use as an aid to reasoning and communication added meaning to the indicative
movements. Those movements served a particular purpose. Technology became a window into
students? thinking as they reasoned and communicated about the representations they were
seeing and with which they were interacting.
In addition to examples previously given, the convergence of ideas can be seen in some
additional examples. Marlon?s illustrated what he was thinking about the function f(x) = x + 1
through the use of indicative movements. His efforts showed that he thought of the idea of x plus
1 in the function f(x) = x + 1 as a movement starting to the left of the origin on the x-axis, going
to (0, 0) and then moving up and to (0,2) and ending at (0,1) as he spoke the words ?x plus one.?
The insight this gives into the misconception of the role of x in the function might not have been
obtained had he not chosen to clarify his thinking through the use of indicative movements. Here
algebraic misconceptions, indicative movements, internal representations, technology as an aid
to mathematical communication, and technology as a window into student thinking interact with
each other.
Another convergence of themes can be seen in work Marjorie did while she was
examining the idea of slope. She used mouse movements to punctuate her explanations,
providing insight into her internal representations. These movements seemed to be motivated by
the request to predict where the next graph was going to land before graphing it. When asked to
make a prediction, she used indicative movements to aid in her explanations. Consider the
following exchange. Figure 9 is provided below for your convenience in following her
explanation.
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Figure 9: This screen shot shows the appearance of the screen at the time Marjorie provided her
explanation of her prediction of where the graph of h
1
(x) = 2x would fall.
INTERVIEWER: Before you hit okay, where do you think . . . two times x is going to be
? show me with your cursor where you think it?s going to land.
MARJORIE: I think 2x might be right here {cursor near (4, 6)} [48:19]. Because the one
and x was right there {cursor at about (6,6) on the graph of v(x) = 1x} that?s the one and
x.
INTERVIEWER: Okay
MARJORIE: And that?s 4 and x {cursor at about (2.5, 10) on the graph of g
1
(x) = 4x}
[48:25] and that?s 3 and x {cursor just below (3, 10) on a segment passing between
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plotted points near f
1
(x) = 3x} so I think that 2 and x will probably be right there {moves
cursor in area of first quadrant between graphs of v(x) = 1x and f
1
(x)=3x back and forth
along a short linear path near where the graph would land}
INTERVIEWER: Okay
MARJORIE: Really because that?s 5, 4, 3,{cursor moves from one graph to the next as
she speaks, hitting the segment when she gets to 3}. I think 2 will be (about) right there
between 3 and 1 {cursor moving in the space in the first quadrant between the functions x
and 3x along a short linear path near where the graph would land}, because I think no
matter what I try or if I do it going this way {cursor in new function window} it?s not
going to put it directly on the dot, the step dot coordinates {cursor near the graph of 3x
and then back to new function window}. Yeah. And I don?t think so. So really 2 (is)
probably a waste of time but I?ll put 2 out there anyway just to see {she clicks okay to
graph it}.
INTERVIEWER: Okay
MARJORIE: Yeah, I was right. {cursor at about (6, 12)} Yeah. 2 was - 2 was just as far
out as 1 {cursor back and forth between 2x and 1x} maybe not as far out but yeah. Its
definitely not {cursor back and forth between 2x and 3x} 2 or 1 definitely not them.
Three is the closest {Cursor near about (5, 15), pointing at the segment joining two of the
data points}.
Note that she purposefully moves the cursor from one example to the next and then
indicates by cursor movements where she thinks the graph of h
1
(x) = 2x will land. Note also
when she is analyzing the results, and saying ?2 was just as far out as 1? that indicative
movements gave clarity to the idea that was in her mind as she said this. It is clear that she knew
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they were two different graphs and that she knew where they were located, whereas her
statement alone would not have provided that information. This example also shows how she
was empowered by the technology. She created different functions based on her own
explorations which helped to test and build on her own understanding. In this example we see
patterning observations, problem solving, reasoning, and sense-making, internal representations,
indicative movements, technology as a window into student thinking, technology as an aid to
mathematical communication, and technology as an aid in the use of standard representations.
In addition to the convergence of ideas present in teaching experiment episodes, it is also
evident that similar influences and uses of technology could be seen in the work of both subjects.
This occurred even though they were working at different mathematical levels. Marlon worked
with a simpler dot pattern. Marjorie had deduced the functional relationship in the more complex
dot pattern. Marlon was confused about the representation associated with a coordinate point.
Nevertheless, both Marlon and Marjorie built increased understanding of what they were
studying and did so in part through their own mathematical choices. Such empowerment arising
from their own choices may help adult learners to grow in their belief in their own effectiveness
as learners of mathematics and as a result improve their achievement (Wadsworth et al., 2007).
In the next chapter, a discussion will be presented as to how the study specifically addressed the
research questions and what it showed about the potential of technology to help build valid,
useful, and enduring internal representations of mathematics.
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5. Discussion
Following a summary of chapters one through three and a description of the limitations of
the study, I will examine the results of this study in light of the research questions. I will then
discuss the implications of the study for adult developmental mathematics students, teachers of
adult developmental mathematics students, the design of developmental mathematics programs,
the general use of technology in mathematics education, and for further research.
Summary of chapters 1-3
The decision to address the problem of the effective use of technology for the
mathematical education of adult learners arose in part from an examination of literature related to
the use of technology in mathematics education, some of which presented a lofty vision of the
potential of such technology use to improve students? learning (Epper & Baker, 2009; Hegedus
& Kaput, 2004; Kaput, 1994). It was also clear that adult developmental mathematics students
form a substantial population that has continuing and unique needs which require the attention of
mathematics educators (Bryk & Treisman, 2010; Epper & Baker, 2009; Parsad & Lewis, 2003).
Using technology in the classroom and doing so wisely has come to be expected of those
aspiring to be excellent teachers of mathematics, and this includes teachers of adult learners
(AMATYC, 2006; National Council of Teachers of Mathematics, 2000). Adult developmental
mathematics students have specific needs that need to be addressed (Epper & Baker, 2009;
Gerlaugh et al., 2007). Technology seems to have particular promise in impacting adult students?
representations (Epper & Baker, 2009; Lapp & John, 2009).
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As the National Council of Teachers of Mathematics has sought to improve the standards
they promote and the resources they provide, appropriate use of representation emerged as one of
the standards for mathematical teaching and thinking (NCTM, 2000). They have also noted
technology?s influence on the role of representation (NCTM, 2000). Some of the ideas examined
with regards to representation have been representational systems, idiosyncratic representations,
visualization, symbolization, the use of multiple representations to describe the same concept,
modeling as a use of representation, and the mathematical idea of functions as a rich context for
examining the use of representations. The ideas of valid, useful and enduring internal
representations emerged as an interpretive framework from a study of the literature related to
representation.
Benefits to the use of technology include the possibility of opening students? ability to
conceptualize mathematical objects and to visualize as mathematicians do (Cuoco &
Goldenberg, 1996; Moreno-Armella et al., 2008). Concerns include the promotion of conceptual
over procedural learning (Fennell & Rowan, 2001). Other issues include the pace of progress in
the cycle of invention, research, and implementation (Fey, 1984; Oncu, Delialioglu, & Brown,
2008). Since the form of technology provided for adult developmental mathematics students is
often computer assisted instruction (CAI), the use of CAI for adult learners is important to
consider. Its benefits include time saved and an increase in confidence level that some students
experience. Challenges include the production of only superficial knowledge and the hindrance
that low literacy skills can be in adult developmental mathematics students? use of CAI (Caverly
et al., 2000; Li & Edmonds, 2005).
Several studies looking at various populations shed light on some of the connections
between technology use and mathematical representations. Software has been developed by
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mathematics education researchers for specific purposes (Abramovich & Ehrlich, 2007;
Yerushalmy & Shternberg, 2001). Studies involving spreadsheets, object oriented programming,
graphing calculators, dynamic geometry software, and technological laboratories also provide
insight. Issues such as the use of multiple representations, self-efficacy, gaining understanding of
the mathematics involved in real-life situations, facing misconceptions, and deepening thinking
are among the issues that research into technology and representation provides (Falcade et al.,
2007; Hennessy et al., 2001; Stevens et al., 2008; Stylianou et al., 2005).
Research on mathematical representation suggests that students sometimes experience
challenges in building their own conceptualizations of mathematical ideas because of the wide
variety of situations, procedures, and symbols present in mathematics (Vernaud, 1998). This is
particularly evident in adult developmental mathematics students, many of whom carry with
them learning disabilities which may make navigating that variety of symbols more challenging,
(Epper & Baker, 2009). Challenges to the implementation of the use of technology in adult
developmental mathematics education also include student discipline and choice as well as
faculty acceptance and perspectives (Epper & Baker, 2009; Kinney & Kinney, 2002). I hoped
that by engaging adult developmental mathematics students in a teaching experiment, ideas
might arise that would help in thinking about how to address some of these challenges. I sought
insight regarding the following questions
1. Following the introductory use of dynamic computer technology to explore
mathematical concepts built upon previous knowledge, what internal
representations of those concepts do developmental mathematics students
possess?
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2. What can be determined about the validity and usefulness of those
representations?
3. How well do those representations endure over a period of time and in the
company of tasks which build upon them?
I decided that the study would be conducted following the qualitative research paradigm
using a constructivist theoretical framework. I developed a teaching experiment in which I could
construct knowledge of the use of technology in adult developmental mathematics education at
the same time that the adult being examined was developing his or her own knowledge at a pace
guided by their own zone of potential construction and my associated movements (Norton &
D'Ambrosio, 2008). I used the ideas of grounded theory as a framework with which to examine
the resulting video, audio, and hard copy data in this case study through the use of a priori, open,
axial, and selective coding.
Limitations
This study is first and foremost limited by its nature, in that it was designed to suggest
ideas for theory and was not intended as a verification of theory. No claims are made otherwise.
In the summaries of the cases and in the conclusions given below, descriptions will be of what
seemed to be happening. Suggestions for follow up will also be included. The study was also
limited in that it is the interpretation of one person. Even though inter-rater reliability test results
may show some consistency in the codes assigned to data, I selected those codes, and my choices
are based on my interpretation of what is essential in the data. As discussed by Eisner (1998), the
view of qualitative intelligence as connoisseurship includes the idea that a researcher?s
knowledge about a situation influences his or her perception of it. In this case, I had a favorable
view of the potential of the use of technology in mathematics education and believed in the
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potential of each student, particularly the participants in the study, to learn from that technology.
This bias may have affected my observations in some way. I also made some interpretations as to
what the subject was thinking. Some of this was validated in by the subjects? own statements, but
it nevertheless leaves room for uncertainty. As final conclusions are drawn, and existing theory
connected, it is hoped that the proper place and power of this study will be clarified.
Conclusions
The specific purpose of this study was to find out how the use of dynamic computer
technology to explore mathematical concepts affected the internal representations of
mathematics possessed by adult developmental mathematics students, as reflected in the research
questions. I hoped that such an investigation would provide information that would allow adult
developmental mathematics educators to make wiser technological choices.
In considering the way in which internal representations might have been determined,
Smith (2003) suggested that he learned more about student thinking through conversations with
his subjects in which they discussed the development of their representations. Goldin (2003) also
noted that research into students? internal representations relies on observations of students?
interactions with and production of external representations and that he proposed that researchers
use task-based interviews. I used teaching interviews in which I held conversations with the
subjects about the development of the technological representations they were using. I also
observed their interactions with and production of technological representations.
When I analyzed the results, the subject?s indicative movements (gestures and mouse
movements), technological choices, written work, and verbal descriptions were all tools which
were available to help suggest student thinking. As noted in the description of cases, the subjects
in this study appeared to make connections between multiple representations. They also appeared
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to find technology to be an aid in the use of standard representations and were in some instances
mathematically empowered through the use of technology. During the course of their work, they
also demonstrated the potential of technology as a means to facilitate communication and to
provide a window into their thinking. In many instances, the insight gained into the subject?s
thinking was revealed through the interplay of their indicative movements and the things they
said about what they were doing as a result of the talk-aloud protocol. Such observation and
conversation is in keeping with the ideas suggested by Smith (2003) and Goldin (2003) for
examining internal representations. A more detailed look at key results of the present study as
related to the research questions follows.
What internal representations followed the use of technology? In considering what
internal representations of mathematical concepts the subjects? possessed, it is first important to
note that the use of dynamic computer technology allowed me to make inferences about those
representations. Observations of indicative movements strengthened inferences about the
student?s internal representations, providing a window into their thinking (Campbell, 2003;
Stevens et al., 2008; Yerushalmy & Shternberg, 2001).
For example, when Marjorie was searching for locations on the coordinate plane where
the x and y coordinates were both the same, indicative movements allowed me to see that she
was restricting her search to the x and y axes. When Marlon decided that the data points he had
graphed to illustrate the mathematics in ?Another dot pattern? was ?just off? of a 45 degree
angle, indicative movements allowed me to see that he considered passing through the origin to
be a requirement of the pattern being exactly at a 45 degree angle. Marlon also revealed his
thinking when he used indicative movements with both paper and technological representations
coincidentally to show that he was connecting the step numbers listed on the paper with the x-
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axis. These and other incidences show that indicative movements can be used to better
understand students? interactions with mathematical representations.
As suggested by the work of Yerushalmy and Sternberg (2001), free movement of the
mouse is important and in the current study the mouse movements used by the subjects of the
study were often not requested for a specific mathematical purpose, but were chosen freely by
the subjects in order to communicate or reason mathematically. The current study also builds on
the work done by Stevens et al. (2008) by demonstrating that a technological medium other than
object oriented programming languages can serve as a ?window into the student?s mind? (p.
199). I extended the work done by Campbell (2003) in examining student?s mouse movements in
that paper and technological representations were connected and indicative movements examined
in both media. I found that indicative movements recorded simultaneously in more than one
media may provide additional insight and relate to and support each other. It also situates the use
of recordings of both the learner and the learners? screen work in a teaching experiment in which
the thinking of adult developmental mathematics students could be studied in depth. Hennessy,
Fung, and Scanlon (2001) in their observations of students? work with graphing calculators noted
the importance of the use of paper and pencil techniques to accompany technology. The current
study adds to this finding by showing additional ways that the two mediums might work
together.
The observations made of the subjects? internal representations showed them connecting
multiple representations and building representations upon their own thinking. Each of these
aspects will be discussed below.
Connecting multiple representations. In considering mental activity many studies have
noted the importance of connecting multiple representations of the same mathematical concept to
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students? understanding (Abramovich & Norton, 2006; Pape, Bel, & Yetkin, 2003; Stylianou et
al., 2005). The subjects of the current study were repeatedly observed to be making connections
between multiple representations. This is demonstrated in Marjorie?s investigations of slope, and
Marlon?s investigations of f(x) = a + b. As seen in figure 9, Marjorie had listed the algebraic
forms of the functions electronically along with the graphs of those functions. She then described
that algebraic information as she moved her mouse along the graphical representations, making
statements such as ?that?s 4 and x? or ?that?s 5, 4, 3? referring to the value of k in f(x) = kx
which went with that particular graph. Marlon was able to see that for functions of the form f(x)
= a + b, he could find the sum a + b and that would tell him where the graph of that function
would cross the y-axis. In this way he was connecting algebraic and graphical representations. In
his final session, when he saw the animated table of values, the algebraic representation of the
function, and the graph of the function, his understanding seemed to deepen as well. He saw the
animated point, saw the different values that point was place in the table, and eventually made
the statement that ?the whole line itself? was being represented.
From this I inferred that the use of technology strengthens connections between
algebraic, graphical, and numerical representations. Additional evidence for this can be seen in
Marjorie?s final session. She found an algebraic representation using technology for points she
had plotted based on a new table of values. She was able to connect that algebraic representation
with the mathematics she was seeing in the table of values, saying ?it is letting me know . . . the
difference between the different points which there is a difference of two add one.? This
statement seems to show that her experiences with the technology strengthened her
understanding of how the algebraic representation related to the numerical information she saw
in the table of values. She was connecting numerical, algebraic, and graphical representations.
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Building representations based on their own thinking. Internal representations which
were being built by the subjects of the study seemed to be based upon their own thinking. This
can be seen in the earlier stages of their explorations when their understanding was weaker.
Marlon?s choice to try f(x) = 1 + 2 to pass through the point (1, 2) was based on his conception
of a coordinate point as a movement to one side and then an upward motion symbolized by
addition. This conception was evident from his mouse movements and written representations of
coordinate points. He moved the mouse to the right and then up to show the locations of
coordinate points. When asked to write down a representation of the point (0, 9), he wrote 0 + 9,
and said, ?It?d be here, zero, plus nine.? In session 4, as he considered how to find a function to
pass through his coordinate points, including the point (1, 2), he noted that he should have said 1
+ 2 and moved the cursor from (0,0) to (1,0) to (1,2). His choice to try f(x) = 1 + 2 as a function
which might pass through the data points for the pattern of dots found in ?Another dot pattern?
(see task in Appendix C) was based upon his idiosyncratic thinking about coordinate points.
Later in the study after he had begun to build some understanding, as shown I asked him
to create a function of his choice. He had noticed that functions of the form f(x) = x + b passed
through the y-axis at (0, b). He had already been working with g(x) = x ? 9. When I asked him to
create a function of his own, he decided to ?add more? and created v(x) = (x ? 9) + 6. He then
observed that the function passed through the y-axis at -3 and was able to deduce correctly that
the -9 and the +6 were combined to give the information as to where the graph crossed the y-
axis. He was building upon his own understanding to understand a function he created.
Marjorie created three different functions which used constant parameters before she
began to build an understanding of the use of x in a function. One of the functions with a
constant parameter which she created was 66)( ??? Bxf . At first she interpreted this
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representation as the quantity 6 plus 6 multiplied by B. When she tried different values for B, she
was able to see that she had been mistaken about the order of operations that should be used, and
saw the operation it represented as B times 6 followed by the addition of 6. This increased
understanding of the order of operations was built on representations she had chosen to create.
Later, she created graphs with different slopes based on the observations she was making about
the effect of the parameter k in functions of the form f(x) = kx. Her choice to try f(x) = 50x was
based on her examination of the table of values she had created for the function she was trying to
graph. When she saw that this function did not come close to her graphed points, she chose to try
different values for k. The understanding she built about the impact of different values of k on
the graphs was built on her choice to examine the representation f(x) = 50x using the technology,
a choice which arose from her own thinking about the table of values.
What was determined about validity and usefulness? Valid representations accurately
reflect the mathematics students seek to reflect and are flexible enough to allow additional
mathematical ideas to be built upon them. They are also accompanied by sound mathematical
habits of mind. Useful representations are accessible for reasoning and sense-making,
communication of mathematical ideas, and building new understanding.
While there was much about the representations they were working with that Marlon and
Marjorie still did not understand, considered within their zone of potential construction (ZPC),
they did build some validity and usefulness in those representations. Marlon?s observation that
functions of the form f(x) = x + b cross the y-axis at the value b added validity to that
representation. It also appeared to become a useful representation for him in his examination of
functions of the form f(x) = (x - a) + b. This is indicated by his examination the y-intercept in
order to understand f(x) = (x ? 9) + 6.
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The valid representation of the graphed points for ?Looking at dot patterns? was useful to
Marjorie in her exploration of the algebraic and graphical representations of that function. She
knew that matching the graphed points was her goal in assessing the validity of the other types of
representations. In addition, her valid conception of the pattern in the table of values included the
understanding that the difference of the number of dots from one step to the next was 3. This
became useful to her. She referred to that idea when she tried f(x) = 3x as a function which might
pass through the graphed points. She observed that three had been the difference between the
number of dots from one step to the next.
The qualities of valid and useful representations can be seen in Marlon?s and Marjorie?s
work. Marlon had built a sound habit of mind to accompany his internal algebraic
representations of functions. Once he saw that the y-intercept correlated with the value b in f(x)
= x + b, he examined the y-intercept in order to understand other representations of functions. In
this way, he used his growing understanding of algebraic representations of functions to build
new understanding. Once Marjorie noticed the pattern in the table of values, this added a habit of
mind to her examinations of other representations. She carried that idea of the rate of change to
her examination of algebraic and graphical representations and to the examination of a different
table of values. When given a new table of values, she noted fairly quickly that the difference in
the y-value from one step to the next was 2. She was able to make sense of this new table of
values and also make sense of the algebraic representation the technology provided for her.
Sound habits of mind, building new understanding, and making sense of things indicate the
validity and usefulness that Marlon and Marjorie were building as they used technology to
examine representations of functions.
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How well did those representations endure over a period of time? The definition
given of enduring internal representations is that they will remain with the student in various
situations apart from the environment in which they were initially developed. They are also
carried forward, built upon, and refined over a period of time. They become part of the student?s
?stored knowledge? (Rogers, 1999). They will become part of the set of mental objects available
to students (Cifarelli, 1998).
Marlon?s mathematical misconceptions interfered with the endurance of what he was
learning. In examining his progress over the course of the study, it was apparent that his
challenges with mathematical vocabulary interfered with the endurance of the verbal
representations needed to use the software without facilitation. He also could not remember how
the function h(x) = x + 1, which represented the dot patterns he had been studying, had been
discovered. He did remember from one session to the next how to change the scale of the graph
and used that feature to solve a problem in the fifth session, leading to his discovery that the
constants a and b in a function of the form f(x) = a + b are ?added together? to give the y-
intercept of that function. In the opening to session 6, when asked to show what he remembered,
he was able to put an xy-plane on the sketch and graphed the function f(x) = 5 ? 9, noting after
observing the plot of the graph that ?they wind up subtracting.? Facilitation helped him to reason
his way back through some of the thinking he had previously done during session 5 and call back
to mind with some meaning the representations he had been studying, particularly the function
which passed through the points representing the dot pattern. Marlon?s handwritten graphs drawn
from his technological work in a previous session were also a help to him in session 5.
Looking at previous work also helped representations endure for Marjorie, who was
aided by referring back to a saved technological sketch. When considering the work done in this
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sketch, Marjorie was able to intelligently discuss the function she had created, namely f(x) = (A -
5) + 20, but could not remember entering the one I had asked her to enter, namely h(x) = x. She
also remembered from one session to next that the scale of the graph could be changed, but not
always how it was done.
These events indicate some possible conclusions about the endurance of internal
representations arising from the use of technology. Saved technological representations which
were created by the student and paper and pencil representations associated with them may help
the student recall previous work more clearly. Dynamic movements of representations may
remain in the student?s mind as a tool for problem solving, even though the technological steps
which produced them do not. Also, enduring representations may be more likely to be built
through tasks in which students have control over their avenues of exploration and which are
situated within their ZPC.
Implications
Though this was a teaching experiment conducted with only two subjects, those subjects
represented a range of developmental mathematics experiences. The suggested theory may apply
to students at more than just one level of developmental instruction. It has implications for adult
developmental mathematics students, developmental mathematics teachers, for those who
manage developmental mathematics programs, for the use of technology in mathematics
education in general, for those interested in understanding student thinking, and for further
research.
Adult developmental mathematics students. There are important implications of this
study that might be shared with adult developmental mathematics students. This study reinforces
the observation that some of the challenges adult learners face is their own lack of self-efficacy,
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lack of study skills, and learning experiences which may be procedurally rather then
conceptually based (Epper & Baker, 2009; Wadsworth et al., 2007). Both subjects in this study
found opportunities to make and check their own conjectures through the use of technology.
Adult learners can consider that some of those investigations would have been difficult for them
to engage in without technology. They can think how this type of empowering investigation may
help them to make valid mathematical learning choices. Such experiences may help build their
self-efficacy. Student centered investigations may also help adult developmental mathematics
students to understand the learning process better in general and thus enhance their learning
skills. In addition, they may see that being able to create and test their own conjectures can help
them learn conceptually and see the advantage of learning conceptually. In the current study, for
example, the subjects sometimes appeared to remember and discuss more clearly their own
investigations.
Adult learners may be encouraged by the progress Marlon and Marorie made over the
course of the study. Both Marlon and Marjorie grew over the course of the study. Marlon began
with great confusion about what he was seeing, and an inability to properly represent coordinate
points, representing them as a sum of the x coordinate plus the y coordinate. He did not
understand what the functional notation the software required meant and saw it as just another
way to graph coordinate points, entering f(x) = a + b in an attempt to graph a line passing
through the point (a, b). During the study he was able to discover that this was a misconception
and that such functions create a horizontal line crossing the y-axis at y = a + b. By the end of the
study, he had learned that functions of the form f(x) = x + b create a diagonal line passing
through the point (0, b), and that such functions pass through other points on the plane. He
learned to associate the step numbers for the table of values representing ?Another dot pattern?
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with the numbers along the x-axis. He also seemed to begin to understand that a function is
associated with points along the entire line, representing ?the whole line itself.?
Early in the study, Marjorie was able to deduce the functional relationship present in
?Looking at dot patterns? merely by her analysis of the table of values representing the step
number and its associated number of dots. She was, however, unable to use algebraic
representations to describe this relationship. During the study she used the software to explore
the xy-plane, learn about the effect of k in functions of the form f(x) = kx and find, recognize,
and describe an algebraic representation of a table of values similar to the one for ?Looking at
dot patterns.? Her description included the meaning of the variable x in that representation. Adult
students can learn from the experiences of Marlon and Marjorie that by using technology in an
interactive manner, they can make their own mathematical choices and investigations, and that
they can understand mathematics at a deeper level than rote memorization.
Teachers of adult developmental mathematics students. Teachers of adult
development students can note that the subjects of this study both brought mathematical
misconceptions to the study which affected their progress. Teachers must do everything they can
to understand the misconceptions adult learners may bring to their classrooms and to the use of
mathematics technology. A better understanding of their students? misconceptions will allow
adult developmental mathematics educators to recognize what their students? zones of potential
construction (ZPCs) are and how they affect their use of technology (Norton & D'Ambrosio,
2008). The findings of this study imply that technology has the potential to help build learning
within adult learners? ZPC.
The results of this study also imply that teachers of adult learners who are incorporating
technology would benefit from having multiple forms of assessment they can draw upon. They
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can consider their students? indicative technological movements and verbal descriptions of what
they are doing. They can also examine their students? use of both paper and technological
representations in order to more clearly assess their understanding. Consider for example that
Marjorie correctly described the algebraic representation for a table of values representing the
function f(x) = 2x + 1 viewing the technology, but when asked to transfer that knowledge to
paper, she used a different representation. When questioned she was able to tell what she meant
by the representation she used on paper and to accurately describe different uses of the variable
that were involved. Such understanding of Marjorie?s thinking would be a great benefit to a
teacher in deciding what step to take next to help build Marjorie?s understanding of algebraic
representations. Validity in making inferences about student understanding should include
adequate relevant evidence (NCTM, 1995). Allowing students to represent mathematics in
different media and asking students to explain those representations may allow teachers to make
more valid inferences as to their students? learning.
Teachers should also take note that Marlon and Marjorie seemed to be able to build more
enduring representations when they had some choice over the avenue of exploration and were
building on those choices they had made. As noted above, teachers should choose tasks in which
students have control over their avenues of exploration and which are situated with those
students? ZPCs. In considering the challenges which adult learners may have in remembering
technological procedures, teachers may also wish to provide clear facilitating tools to accompany
the use of technology so that the dynamic representations and the empowerment they provide can
be activated by the student for learning as easily as possible.
Design of developmental mathematics programs. Research suggests that
developmental mathematics programs may employ teachers who lack appropriate professional
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development or have little experience dealing with the special needs of adult learners (Caverly et
al., 2000). The teachers may also have beliefs which adversely affect the technological choices
they make for their students (Caverly et al., 2000). Developmental mathematics programs should
attend to the potential advantages of the appropriate use of technology which this study
highlights, such as the mathematical empowerment of adult learners. Program directors may
wish to make certain that dynamic interactive software such as Geometer?s Sketchpad is
available to their teachers and students and that their teachers receive some training in how to
implement such technology into their classrooms. With appropriate training, teachers may be
able to take greater advantage of time spent in computer laboratories as they learn to select
appropriate tasks and assess their students? understanding. They can be trained to make valid
assessments as they observe their students? interactions with technology.
They can also learn to question their students in ways that elicit valid information as they
circulate among and interact with them. Teachers can learn to make appropriate inferences from
the indicative movements they observe and the students? talk-aloud descriptions of their own
work. In so doing, teachers of adult learners may be able to more easily uncover some of the
misconceptions that their students possess, giving them greater power in serving those students
(Li & Edmonds, 2005).
Directors of developmental mathematics programs should also note that there are
indications in this study that adult learners may not be carrying into their current coursework the
knowledge they are expected to have. Marlon was enrolled in the second level developmental
mathematics course. According to the course description for the first level developmental
mathematics course, which he had recently passed, he should have been introduced to algebra in
a manner that included a discussion of linear functions, but he began the study with a lack of
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understanding of the representations associated with linear functions. Marjorie was in the third
and highest level of developmental mathematics. It is not known whether or not she recently took
the previous course, but her placement in the highest level course implied that she was expected
to have greater knowledge of algebraic representations than she seemed to possess. The second
level of developmental mathematics included basic algebra, linear and quadratic functions and
she should have had that foundation, but had difficulty expressing ideas algebraically. Those
who direct developmental mathematics programs may wish to examine the knowledge their
students are carrying into subsequent levels of instruction, and examine the effectiveness of their
programs. It may be that alternative solutions which include the use of dynamic mathematics
technology would provide more enduring representations of mathematics and more lasting
conceptual knowledge for these students.
The general use of technology in mathematics education. The use of technology as a
window into student thinking, as an aid in the use of standard representations, as an aid to
reasoning, as an aid to mathematical communication, to reveal and clear up misconceptions, and
to empower students mathematically need not be restricted to the education of adult
developmental mathematics students. These influences and uses may well occur with other
populations, and teachers at all levels may wish to investigate how their own use of technology
in the classroom may be expanded to include any or all of these uses not currently being realized.
Middle school or secondary mathematics teachers, for example, may wish to allow their students
to explain mathematical concepts using dynamic graphing software and indicative movements to
show their thinking.
The power that is found in these potential effects of the use of technology is much greater
than is often found in the ways that technology is actually used. In the present study, a dynamic
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interactive form of technology was used and the subjects made their own choices about their use
of standard representations, receiving rapid feedback as to the effect of those choices. This
feedback was embedded in multiple connected representations, sometimes dynamically animated
so that the effect of changes in the representations could be seen. Such technology empowered
these subjects in ways that some other forms of technology would not have been able to do.
Forms of computer aided instruction (CAI) which do nothing more than provide electronic
textbooks or multiple choice question and answer sessions may have their place in reinforcing
what students have already learned, but would probably not provide the kind of cognitive power
that dynamic connected interactive representations of equations, tables, and graphs was able to
give these adult learners.
Further research. The current study opens up many avenues to potential further research
(Epper & Baker, 2009). Potential areas of continuing research are the use of technological
gestures, the interplay between previously held misconceptions and technological
representations, the effect of technology on the internal representations of different populations
of students, the use of indicative movements in a large group setting, the use of paper and pencil
in combination with technology, and longitudinal work into the effects of dynamic mathematics
technology in fostering enduring representations.
Additional research into the use of technological gestures may be able to provide
teachers with practical ways to record and analyze such work. It may give them further support
in using such representations to understand their student?s thinking. Such gestures could be
observed in various technological settings in addition to the one used here.
Research into how adult students? misconceptions interfere with their use of technology
can provide teachers with information about how to facilitate the use of technology. Such
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research might show teachers how to diagnose difficulties students are having by observing
particular technological behaviors. This might allow teachers to discover weaknesses they had
not suspected, giving mathematics technology use additional power.
Future studies may include a similar examination of indicative movements and the
resulting insight into student thinking that might be obtained with different populations of
students. These might include elementary, middle, or secondary school students or college level
students who do not require remediation. An examination of such populations may help
generalize the influences of uses of technology noted. Younger students may also become more
easily acquainted with new technologies. Observing indicative movements may give teachers at
every level insight into their students? misconceptions, reasoning, and thinking. It may also
provide students of all ages with a means to communicate mathematically which they had not
previously considered. Teachers at all levels could conduct similar experiments with their own
students in after school settings.
Campbell (2003) conducted his study in dynamic tracking in a large group setting with
elementary pre-service teachers. Large group research settings are also possible with the methods
used in the current study. Screen capture software could be installed in laboratory computers, and
computer cameras used to record the subjects? expressions and actions. Students? paper artifacts
could be recorded as well using additional small cameras or collected for examination. Observers
could circulate and question the subjects and their field notes could be used as a basis for further
examination and analysis of particular subjects? interactions.
In the current study, I presented the subjects with a paper and pencil activity on which
the technological activity was built. During the course of the study, paper and pencil and
technological interactions were both used, sometimes simultaneously, as when Marlon was
209
connecting the step numbers on paper with the x-axis represented on screen. Marjorie wrote her
thoughts about patterns represented by a table in her own words on paper and then connected
those representations with technological ones. A subject of further study might be the
interactions of paper and technological representations. How would the study have been different
if the subjects had been presented with technological representations first and then asked to use
paper and pencil to assist them in their investigation? How can teachers make decisions about the
use of paper and pencil together with technology?
One of the ideas presented in this study was the possible endurance of technological
representations beyond the use of technology. Because of the limited nature of the study,
examination of endurance was limited. Further longitudinal studies could be conducted which
look more closely at what students retain from dynamic interactive technological tasks designed
to build conceptual understanding. For example, if such tasks were used in the first level of
developmental mathematics, would students then retain more of what they had studied as they
moved through the second level of developmental mathematics? Would summative paper and
pencil assessments indicate that any of the internal representations they may have gained through
the use of technology had endured with validity and usefulness?
Conclusion
Technology has the potential to empower adult developmental mathematics students to
strengthen their internal representations of mathematics, allowing those representations to grow
in validity and usefulness. Technology appears to have the potential to allow adult learners to
build understanding by choosing their own avenues of exploration situated within their zone of
potential construction (ZPC), although students may also bring misconceptions to these
activities. Teachers seeking to provide such empowerment must first carefully consider the
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misconceptions related to the use of technology. Teachers need to keep these misconceptions in
mind when assessing technological interactions. Such misconceptions as well as other aspects of
student thinking may be revealed by indicative movements. Teachers must be ready to provide
clear technical facilitation so that learners will be empowered by the capabilities of technology
and not hampered by difficult technological procedures. Teachers may also benefit from making
assessment inferences using multiple sources of data, not necessarily relying on technological
interactions alone.
Marlon?s cry of ?Ah!? after he had chosen to graph v(x) = (x ? 9) + 6, noticed where it
crossed the y-axis, and connected that graph with the algebraic representation and his prior
understanding is indicative of the empowerment technology can give adult learners. Such an
exploration may have been difficult for him to do without the use of technology. There was still
much he did not understand, but he did understand something more than he had previously
understood. He made his own mathematical choices and used his own reasoning to analyze the
results. Coming from a setting in which he was failing at his attempts to remember algorithms,
this was mathematical empowerment. Technology has the potential to prevent developmental
mathematics from being the ?insurmountable barrier? it was recently described as being by Bryk
and Treisman (2010), who noted that difficulties in developmental mathematics was ?ending . . .
aspirations for higher education? (p. 19). In the year 2008, a higher percentage of African
Americans ages 25 to 34 were enrolled in some kind of schooling than were Whites, Hispanics,
or the total population (Snyder & Dillow, 2010). Could they have been trying to make up for a
lack of opportunity to learn earlier in their lives similar to that described by Tate (1995) in his
discussion of African American students? experiences in urban schools? Adult developmental
mathematics students need not be victims of traditional teaching methods which have been
211
passed down as an artifact, but which have not been effective in meeting their learning needs
(Tate, 1995). Proper use of technology can change adult developmental mathematics education
for the better. Considering the efforts put forth by adult learners, the special needs they have
which have affected their opportunities to learn, and the substantial population whose futures
may be at stake, mathematics educators cannot afford to ignore this work.
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Appendix A
Glossary of Terms
229
Glossary
Word Definition
Affective domain ?The affective domain refers to feelings that pertain to mathematics, to
the experiencing of mathematics, or to oneself in relation to mathematics?
(Goldin, 2003, p. 280)
Behaviorist Behaviorist teaching models may provide a stimulus and response
approach without regard to conceptual understanding.
Computer
assisted
instruction
The use of software designed to tutor students, provide them with extra
practice, and sometimes engage them in dialog is sometimes known as
computer-assisted instruction (CAI) (Kinney & Kinney, 2002).
Concentration Obiekwe (2000) in his discussion of the instrument used by Wadsworth et
al. (2007) noted that concentration was thought of as a student?s ability to
give attention to an academic task.
Conceptual
knowledge
Knowledge that is part of a network of connections to other ideas (Kinney
& Kinney, 2002)
Constructivism Based on ideas of Piaget and Vygotsky, constructivism encouraged
educators to create an atmosphere where students could work through
cognitive conflict using their own strategies, and thus learning via
problem solving (Lambdin & Walcott, 2007). Students then have the
opportunity to construct their own knowledge (Silver, 1990).
Covariation Considering functions from a standpoint of covariation means that
variations in the independent variable (or input) and dependent variable
(or output) are considered together.
Developmental
education
Developmental education, sometimes referred to as remedial education,
refers to educational efforts which serve college students who need
additional preparation in order to be successful (Payne & Lyman, 1996)
Direct instruction Wadsworth et al., (2007) seem to use this term to indicate in-classroom
instruction with a teacher present, as opposed to computer instruction
students engage in independently outside of class.
Embodied,
linguistic,
formal, and
internal
representations
Embodied representations of mathematical ideas are external, physical
situations in the environment.
Linguistic representations are those in which the emphasis is on syntax
and semantics.
Formal systems use symbols, axioms, definitions, constructs, etc.
Internal, individual systems describe thinking processes and are inferred
from behavior or introspection (Goldin & Janvier,1998)
230
Word Definition
Formative
assessment
Assessment designed primarily to provide constructive feedback to the
student so that he or she may improve
Function A mathematical relationship in which one set of data is matched with
another set of data so that each piece of data in the input set is matched to
one and only one piece of data in the output set.
Function
mappings
A function mapping diagram renders the x and y axes as parallel lines,
with line segments connecting a point on the x-axis to its image on the y-
axis. (Bridger, 2001)
Global To deal with the function in a global way is to look at its overall behavior,
such as when students sketch the graph of a function and look at its
maximums and minimums and other characteristics. (Even, 1998)
Idiosyncratic
representations
Those which are unique to the learner (Smith, 2003). Such personal forms
of representation, which may be very meaningful to the student, but have
little resemblance to those commonly used
Information
processing
A student?s ability to process knowledge. (Obiekwe, 2000)
Internal and
External
representations
The words external and internal refer to the relationship of that
representation to the mind of the student. If the representation exists
within the mind of the student, then it is an internal representation. If the
representation is found in the environment outside of the student?s mind,
in a textbook, on a computer screen, or on a piece of paper for example,
then it is considered to be an external representation. (Goldin, 2003)
Isomorphism A one to one relationship between two sets of data preserving operations
within the two sets. (Dictionary.com, 2010)
Manipulatives Hand-held objects used to model mathematical ideas.
Model A mathematical model is a form of representation which illustrates
mathematical features of a complex phenomenon and is used to clarify
situations and solve problems. (NCTM, 2000)
Non-cognitive
factors
Non-cognitive factors include influences unrelated to the student?s
knowledge and may include such items as motivation and time
management. (Gerlaugh, 2007)
Object oriented
programming
As used in mathematics education, software which provides powerful
tools such as drawing implements and other graphics in a setting
requiring relatively simple syntaxes with which students can create their
own programs. (Connell, 1998)
231
Word Definition
Pointwise To deal with functions in a pointwise way is to plot, read or deal with
discrete points. (Even, 1998)
Reference field A set of objects related to a set of representations. For example, one
reference field for the set of nouns is the set of material objects. (Moreno-
Armella et al., 2008)
Reification The process by which something abstract becomes real to the learner and
exists in his or her mind as a mental object. (O'Callaghan, 1998)
Representation This term refers to both ?the act of capturing a mathematical concept or
relationship in some form? and ?the form itself? (NCTM, 2000, p. 66).
Cuoco and Curcio (2001), described a representation as a map of
correspondence between a mathematical structure and a better understood
structure.
Self-efficacy A person?s belief in their own ability to be effective in managing future
situations. (Wadsworth, Husman, Duggan, and Pennington, (2007)
Self-testing A student?s ability to prepare for tests and classes. (Obiekwe, 2000)
Semiotic
mediation
The process by which new signs are derived from the actions performed
with another sign or symbol. It indicates an internalization process,
producing a new internal tool. (Falcde, Laborde, & Mariotti, 2007)
Semiotics Semiotics is the study of signs and their meanings. Cunningham (1992)
described semiotics as "a way of thinking about the mind, and how we
come to know and communicate knowledge" (p. 166).
Symbolization The process by which symbolic structures re-design the architecture of
the human mind and provide a meta-cognitive mirror in which our
thought is reflected. (Moreno-Armella et al., 2008)
Technological
representations
Charts, graphs, geometric shapes, algebraic equations, or other
mathematical objects represented via mathematics software, internet sites,
or on hand-held devices such as calculators.
Textual
(descriptive) and
visual (depictive)
Textual representations are semantically dense, and conveyed through
rules. Visual representations are more analogical. (Sedig, 2008)
Thick description
Thin description would merely describe the action. Thick description
would give more than just a description of the action. It might provide,
for example, information regarding the motivation for the action. (Geertz,
1994)
232
Word Definition
Title IV Title IV institutions are those which participate in certain federal student
aid programs. (Aud et al., 2010)
Visualization The creation of a mental image to guide the representation of ideas.
(Presmeg, 2006)
233
Appendix B
Interview Protocols
234
Interview Protocol: Initial Interview
Project: the Effect of Technological Representations of Developmental Mathematics
Students? Understanding of Functions
Time of interview:
Date:
Place:
Interviewer:
Interviewee:
Obtaining of informed consent: Before anything else, I need to obtain your signature on
this document. It officially lets you know about the study and what your rights
and obligations are. (point to first paragraph). You can read here that the purpose
of this project is to find out how using computer software to study mathematics
affects how a person thinks about mathematics. Look over the rest of the
document and let me know if you have any questions. (Answer questions as
needed, obtain signatures.)
Obtaining personal information: Ask subject to fill out the subject information sheet.
Further description: This first interview will help me get to know you a little bit and find
out what your experiences with math have been. I?ll be selecting four people to
continue on and participate in more interviews where we?ll look at some math
computer software and I can see how they think. No one will have to pass any
kind of a test to be selected to continue. I?ll be picking people to continue based
on how they will fit in with what I want to learn and what I?m doing to do to learn
it. Do you have any questions about the project?
Questions and Prompts:
1. Tell some things you remember about school when you were growing up.
2. Describe a (another) good experience you remember having in school.
3. (If not already stated) What helped you learn?
4. (If not already stated) What made learning harder?
5. Talk to me about your most memorable teacher.
a. How did they teach?
b. How did you feel about that class?
6. (If not already mentioned) Describe your most memorable math teacher.
a. (If not already stated) How did they teach?
b. (If not already stated) How did you feel about that class?
7. What about your other math classes?
235
8. What other things do you remember about your math classes?
9. Did your teachers ever use any kind of technology in teaching mathematics?
a. That includes calculators, computers, and the internet . . .
b. (If so) What did they use?
c. How did they use it?
d. How did you feel about using ________________ to study mathematics?
e. Do you think ___________________ helped you learn?
10. (Give them a blank sheet of paper and a pencil) Show me and tell me about some
mathematics you remember. It can be anything that you remember.
a. Why did you pick this to share?
b. Additional probing questions might focus on the mathematics they share,
but not for the purpose of measuring achievement, e.g.
i. Can you tell me anything else about ___________________ ?
ii. NOT What?s the answer to _________________ ?
11. I?m going to give you something to look at now, and I want you to just tell me
everything you can about it. Don?t worry about what the right answer is, or what
math you?re supposed to use or anything like that. Just look at this, read the
directions, and follow them. One of the other things I want you to do is talk out
loud about what you are doing as you work, talking continually, as if you were
thinking out loud and I were listening in on your thoughts. (Give them ?Looking
at patterns).
a. Probing questions for this and number 12 might include:
i. How will the pattern continue? Can you draw the next shapes in
the pattern?
ii. What will the 20
th
shape in the pattern be?
iii. Can you tell what the 35
th
shape in the pattern will be (without
drawing it?) What about the 41
st
? How do you know?
12. (If this seems not to challenge them, you may continue with one or more of the
following) Here is another idea for you to think about. (Give them ?Looking at dot
patterns? and/or?Soda cans?). Read the directions and follow them. Talk out
loud about what you are doing as you work, talking continually, as if you were
thinking out loud.
a. What will the next pattern be?
b. Can you tell me how many dots the 10
th
pattern will have? How?
13. Thank you for participating. Is there anything else you?d like to say?
236
Teaching experiment interview protocol: Exploring patterns
Project: the Effect of Technological Representations on Developmental Mathematics
Students? Understanding of Functions
Time of interview:
Date:
Place:
Interviewer:
Interviewee:
Questions and prompts:
Recalling the last session and examining the data:
1. What do you remember about this dot pattern that you saw last time?
a. Follow with probing questions, such as those listed below, to elicit student
thinking
i. Why did you think that?
ii. Is there anything else you?ve noticed about the pattern?
iii. How do you know what the number of dots in the next pattern
will be?
1. Is there a way you can write down this information?
iv. Do you know what a table of values is?
1. If they do: Show me what you know about tables by
creating a table of values showing what you know about the
step numbers and numbers of dots.
a. Tell me about your table.
2. If they don?t or are unsure: then go ahead and give them
page 1 of the exploration, so they can see the blank table.
a. Does this look familiar? What do the columns tell
us?
b. Enter what you know about the step numbers and
number of dots.
2. Look at the information in your table and see if you notice any patterns.
Record your thoughts and talk about them as you write.
a. Subject may also compare back to the dot pattern representation.
b. What is the number of dots in the _____ th pattern? (several steps beyond
the data they have).
c. Can you give me a rule for how many dots would be in any pattern?
237
i. If they can, show them another pattern until you find one for
which they can?t. Finding the rule will be the overall question
this teaching experiment seeks to answer.
Introduction to the software
3. We are going to use mathematical software to study this question. The name
of the software is the Geometer?s Sketchpad. (Open the software) Have you
ever heard of it?
a. (If so) Open a new file and show me what you know about the Geometer?s
Sketchpad.
b. (If not) We?re going to explore it. Open a new file. Make it as large as you
can.
i. What do you notice?
ii. Hold your cursor over the buttons at the left.
1. What do you notice?
2. What are the names of the buttons?
3. What happens when you hold the buttons down?
iii. Try some of these buttons and tell me what you notice.
c. Open a new file. From the menu at the top, select Graph, define
coordinate system.
i. Does this remind you of anything?
ii. (If so) What is it? What can you tell me about it?
iii. (If not, unsure, or more scaffolding is needed regarding the xy
plane) Use the point tool to put a point somewhere in the
coordinate system. Use the measure menu, and measure the
coordinates of the point. Use the selection arrow tool and move
the point around. Talk about what you see. What are the
coordinates telling you?
1. (If needed to elicit further observations). Use the graph
menu and choose snap points if you want the point to
always land at an intersection. Look at the coordinates.
a. What do you notice as you move the point around?
Try this for a minute and as you do so, talk about
what you see.
2. Put the point at a place where
8
a. Both coordinates are the same
b. The right is twice as much as the left
c. The right is half as much as the left
8
As suggested in Key Curriculum Press (2002)
238
d. The right equals the negative of the left or the left
equals the negative of the right
d. Using the selection arrow tool, select the point at the middle of the crossed
axes (called the origin) and move it to see how the coordinate grid moves
around.
i. What?s happening to the point you put on the grid?
ii. What about its coordinates?
e. Using the selection arrow tool, select the other point you see, the one at 1
on the x-axis and move it to see how the grid changes.
i. Talk about what you see.
f. Using graph, plot points, enter some coordinates of your choice and see
where those points lie. See if you can predict where they will be.
i. Move these points around. How do they behave differently from
the other points you have put on the graph?
Using the software to look at their data:
4. How can you use these tools to explore your data?
a. What data do you have?
b. What value in your data could be the left coordinate of a point? What
could be the right coordinate to go with that point? Why?
5. If they have graphed their points: What pattern do your points form?
a. (If they think the points are in a straight line). How could you find out for
certain?
6. What if you had another step in the sequence of dot patterns, whose data is not
already listed in your table? Where do you think the point representing that
data would fall?
Additional questions may be asked, based on the student?s thinking.
239
Teaching experiment interview protocol: Exploring functions
Project: the Effect of Technological Representations on Developmental Mathematics
Students? Understanding of Functions
Time of interview:
Date:
Place:
Interviewer:
Interviewee:
Questions and prompts:
Recalling the last session and starting algebraic notation
1) Begin with GSP in its opening configuration
a. Show me what you remember from last time. Talk about what you see
i. What is the problem we?re trying to solve?
ii. What do you know about that problem?
iii. What have we done so far to explore that problem with GSP?
iv. What does the table tell us?
v. What does the graph tell us?
vi. If I asked you now how many dots were in a certain step number,
how would you find out?
vii. If we had a rule, how would we describe it?
1. Look at a simpler pattern for which a rule can be found and
help the subject to see how to use algebraic notation to
describe it.
240
a. How is the number of dots related to the step
number?
b. Can you write that in a sentence?
c. How can we write that in a simpler way?
i. Use boxes to represent the step number if
scaffolding to the use of variables is needed.
Using the software to explore algebraic notation
2) Go to graph, new function. In the rectangle, you can enter expressions, or rules.
a. Enter the rule we found for our pattern
b. Hit okay.
c. Hit plot function.
d. What do you notice?
e. Why does the graph look the way it does?
i. Explore some different expressions. You may change the colors to
match using the display, color menu.
f. What patterns do you notice?
Additional questions may be asked, based on the student?s thinking.
241
Teaching experiment interview protocol: Exploring parameters
Project: the Effect of Technological Representations on Developmental Mathematics
Students? Understanding of Functions
Time of interview:
Date:
Place:
Interviewer:
Interviewee:
Questions and prompts:
Recalling the last session
3) Begin with GSP in its opening configuration
a. Show me what you remember from last time and talk about what you are
doing.
b. What did you find out about how the graphs of different equations
behave?
c. What is different about their . . .?
i. Graphs
ii. Tables
iii. Equations
Using the software to explore parameters
4) Go to graph, new parameter. Enter a letter name (other than x or y) for the new
parameter to replace the one that is there and hit okay.
5) Define and plot a new function which uses that parameter in its equation (find it
under ?values?).
6) Double click on the parameter and change its value. Before you hit okay, predict
how you think the graph will change.
a. Try this until you can describe in general how this parameter affects this
type of function.
7) Select the parameter and edit, action button, animate parameter. Predict how you
think the graph will change as the parameter changes. Hit animate parameter and
see what happens.
a. Use the animation feature of GSP to further explore the equations of
functions. Talk about what you are noticing.
8) What have you learned about how equations of functions behave?
Additional questions may be asked, based on the student?s thinking and his or her
progress to this point.
242
Teaching experiment interview protocol: Exploring applications
Project: the Effect of Technological Representations on Developmental Mathematics
Students? Understanding of Functions
Time of interview:
Date:
Place:
Interviewer:
Interviewee:
Questions and prompts:
Recalling the last session
9) Begin with GSP in its opening configuration
a. Show me what you remember from last time and talk about what you are
doing.
b. What did you find out about how parameters affect the way different
functions behave?
c. What have you learned about their . . .?
i. Graphs
ii. Tables
iii. Equations
Using the software to explore parameters
10) Use the things you have learned so far to explore the problems on this handout.
Talk about what you are thinking and doing as you explore.
Additional questions may be asked, based on the student?s thinking and his or her
progress to this point.
243
Appendix C
Tasks
244
Looking at patterns
Study the pattern below and tell me everything you notice about it.
245
Looking at dot patterns
Study the pattern below and tell me everything you notice about it.
246
Soda Cans
How many soda cans will you put in the next stack if it follows the same pattern? Draw a
picture to show the stack. What else can you say about this pattern? What if the manager
of the store wanted 10 cans across the bottom of the stack? How many total cans would
be in the stack?
247
Another dot pattern
Draw the next pattern in the sequence.
How many dots are in each pattern?
What else do you notice about the patterns?
Fill in the table below for the patterns
How many dots will be in the
10
th
pattern?
25
th
pattern?
107
th
pattern?
How do you know that?
If I let x represent the step number, how many dots will be in the x
th
pattern? Why?
Step
number
Number
of dots
248
Exploring patterns
Enter what you know about the step numbers and number of
dots for the pattern you are analyzing.
Look at the information and see if you notice any patterns.
Record your thoughts below.
Step
number
Number
of dots
249
Use the grid and space below to record what you noticed about the things you explored
today
9
.
9
Graph image found at http://faculty.matcmadison.edu/kmirus/GraphPaper20x20AxesUnits.bmp
250
Exploring Parameters
10
Change the values of some of the parameters in a function you have used and observe
what happens. Make notes about what you find.
Notes about the effect of parameters:
10
Graph image found at http://faculty.matcmadison.edu/kmirus/GraphPaper20x20AxesUnits.bmp
251
Do you think those parameters would have the same effect on other functions? How
could you find out?
Use technology to investigate the effects of changing parameters on different types of
functions.
Notes about the effect of parameters:
252
Notes about the effect of parameters:
253
Notes about the effect of parameters:
254
Summarize what you have noticed about the effect of parameters on functions:
255
Exploring functions
How does the equation for a function affect the way a function looks?
Part 1: Exploring equations and graphs
Use the Geometer?s sketchpad to explore different equations of function. Make a record
of your work by sketching a graph in each grid below and labeling the graph with the
equation that goes with it.
11
________f(x) =______________________ _____ f(x) =______________________
________f(x) =______________________ _____ f(x) =______________________
11
Graph image found at http://faculty.matcmadison.edu/kmirus/GraphPaper20x20AxesUnits.bmp
256
________f(x) =______________________ _____ f(x) =______________________
________f(x) =______________________ _____ f(x) =______________________
What did you find out about how the equations affect the graphs?
257
Part 2: Find an equation to go with points
Give the table a title and headings and record coordinate points in the table. Graph those
points in the Geometer?s Sketchpad and below. Use the Geometer?s Sketchpad to find a
function which will pass through most or all of your points. Record the equation and
graph for that function.
________________
________f(x) =______________________
How did you find the best function?
258
Exploring Different Applications of Functions
Part 1: Data for Albert and Carl
Open the sketch ?Race with adjustable paths.? Animate the points and see what happens.
Drag the points back to the start.
1. Have Geometer?s Sketchpad create a table of values which will keep track of
Albert and Carl?s distance from the finish line, updating the table every 10
seconds.
2. Run the race again.
a. Where was Carl at 4 seconds?
_________ ___________________________
b. Where was Albert at 4 seconds?
____________________________________________
3. Use the table you see in the Geometer?s Sketchpad to fill in the table below.
4. Use what you have learned in the Geometer?s sketchpad to find one function that
shows us Carl?s movement, and one function that shows us Albert?s movement.
You may open a new sketchpad file in which to work.
5. Sketch both graphs on a paper grid and label the grid
12
.
12
Graph image found at http://faculty.matcmadison.edu/kmirus/GraphPaper20x20AxesUnits.bmp
Seconds Carl?s distance from finish Seconds Albert?s distance from finish
0 0
1 1
2 2
3 3
4 4
5 5
6 6
7 7
8 8
9 9
259
6. What can you tell me about Albert?s and Carl?s movement? How do their
equations and graphs compare?
260
Part 2: Adjusting the Race
1. Drag Albert and Carl back to the start. Make an adjustment in the sketch so that
Albert and Carl will finish at the same time.
2. Set up a table as you did before and run the race again so you can keep track of
this data, showing where they are as each second passes.
Seconds Carl?s distance from finish Seconds Albert?s distance from finish
0 0
1 1
2 2
3 3
4 4
5 5
6 6
7 7
8 8
9 9
261
3. Find the functions to go with their new race and graph them. Sketch the graph on
a grid and label the grid.
262
4. Describe how you adjusted the race and why and describe how you found the
functions.
5. How do these equations and graphs compare to each other?
6. How do they compare to the equations and graphs for the first race?
263
Appendix D
Coding Guide
264
Coding Guide
The purpose of this guide is to explain the method that was used to unitize and
code the transcriptions of both the initial interview and teaching experiment sessions held
during the course of this study. Feeds from the computer screen, the work done on paper,
and a view of participants were all coordinated together for the final video recording.
Transcriptions were made of these video taped sessions. Notations were made of timing
and actions of the participant in addition to what was said.
Unitizing
The table below shows the definitions used to describe units of analysis for this study.
Term Definition
Topic A complete idea, type of technological action, or topic
of conversation upon which the attention of the speaker
or speakers is focused.
Unit A word, sentence, paragraph or several consecutive
sentences or paragraphs focusing on the same topic.
Examples of Unitizing
Dividing a section into units
Below is a section of dialog which contains more than one unit of analysis
INTERVIEWER: Okay. Could you tell me - can you tell me what the 35th shape
in the pattern will be?
MARJORIE: okay, 20, 22, 24, 26, 28 {she circulates back in her counting to the
beginning of what she has illustrated} 30, 32, 34, a rectangle, 35.
INTERVIEWER: Okay, and how did you know that? [31:17]
MARJORIE: I just did the twos, I counted by twos.
INTERVIEWER: Alright. What about the 41st? [31:28]
MARJORIE: 41st, Okay so if that?s {gesturing as she counts across the shapes}
35, 36, 37, 38, 39, 40, 41, triangle. And I counted that with, just counted one
{laughs}
This section of dialog would be unitized as follows. Notice that each unit begins with a
key word and a question on the part of the interviewer. Because finding the 41st pattern
265
in a pattern which was five shapes long was qualitatively different from finding the 35th
pattern, this question was considered to have started a new topic.
First unit INTERVIEWER: Okay. Could you tell me - can you tell me what the
35th shape in the pattern will be?
MARJORIE: okay, 20, 22, 24, 26, 28 {she circulates back in her
counting to the beginning of what she has illustrated} 30, 32, 34, a
rectangle, 35.
INTERVIEWER: Okay, and how did you know that? [31:17]
MARJORIE: I just did the twos, I counted by twos.
Second unit INTERVIEWER: Alright. What about the 41st? [31:28]
MARJORIE: 41st, Okay so if that?s {gesturing as she counts across
the shapes} 35, 36, 37, 38, 39, 40, 41, triangle. And I counted that
with, just counted one {laughs}
266
Keeping a section in one unit
Below is a section of dialog which was kept as one unit. The rationale for doing so is also
given. The boldfaced sentence might indicate a change of topics, but in this case I felt it
was important to consider this passage as one unit.
Unit Rationale
INTERVIEWER: Okay, so can you write on your paper what how you (would)
write that in an algebraic expression? Can you write that down?
MARJORIE: {writes y = 5 x + 1} but since I know what x is , then
INTERVIEWER: What do you mean - what is x?
MARJORIE: fi- x is two {writes y = 5(2) + 1 = 11} () gives you ll
INTERVIEWER: So what would the equation be for, for this step
MARJORIE: For step number three, it?d be the exact same thing you?re just
replacing the five with three. (It) would be three x plus one which would (get
you) 7 and then since you know that the difference is two, x is two. 3 times 2 is
6 plus one equals 7.
INTERVIEWER: Okay -
MARJORIE: Hmm, mm hmm. Because when I first saw it, I was, I mean the
one thing I did notice when I first saw it was that it was odd numbers {she
gestures with the pen down the #dots column of the table} and I was like, oh,
okay (something) there. But then once I started looking at it, and trying to you
know, figure out what the difference was {she gestures back and forth from left
to right entries in the table} because no matter what there?s always, there?s
always got to be well not always but there seems to be a difference you know
as far as these numbers just aren?t randomly {gesturing down the #dots
column} picked especially since I?ve been in this study. These numbers are not
just randomly picked {some gesturing back and forth again as she speaks,
INTERVIEWER: laughs} so its got to be a pattern and I just started you know,
looking down the different steps till like {gesturing with pen down the step
column}- okay and since um, and then when I didn?t, when I couldn?t see a
correlation between the step and the dots, {gestures from left to right across
table} I started looking at the dots {gestures with pen down the #dots column}
because the steps were just going you know straight down {gestures down the
step column} just like it was before. When I was looking I was like the dots
{gestures down the #dots column}- what?s the difference between the dots and
then I just started counting and I got you know difference (from) when you add
two to three you can get five if you added five, if you add two to five you get 7,
and so forth {waves pen in air above the table}. I just went down. And then I
noticed that pattern of two,{some gesturing towards #dots column} it was
before it was three, but this one here is two.
I wanted to understand
and I wanted her to think
about her reasoning for
saying 5(2) + 1 = 11 so
the highlighted question
is a continuation of the
same topic: her reasoning
for her paper
representation being
different from the
representation given by
the software. Note that
following the boldfaced
statement, she says ?it?d
be the exact same thing.?
267
Coding
This study was entered into with a handful of a-priori codes, some of which
evolved over the course of the study and were divided into codes which captured the
nuances more clearly. Though specific names for codes may have changed over the
course of the study, those listed as a-priori below reflect the ideas looked for going into
the study. Some emergent codes were given names which were associated with the
literature. Other unexpected ideas arose as well. Codes referring to characteristics which
are beyond the scope of this study are not listed below.
Code Family: Mathematical Thinking Processes
Emergent codes
Code Description Example
Math-AM: Algebraic
misconceptions
Quotations where it is evident that
the subject doesn't understand the
role, meaning, or purpose of an
algebraic representation as
commonly used or as related to
other mathematical ideas.
?Okay, x + 9 so the x is equal to
9?
Math-FC: Function and
Coordinate point confusion
Quotes where the representations
and concepts related to functions
and coordinate points seem to be
becoming confused
?Do you remember the different
ways you plotted those points?
MARLON: Maybe, plot new
function??
Math-GC: Graphical
confusion
Quotations where the subject
exhibits Confusion about how
points are located on the
coordinate plane, the terminology
used to describe them, or the
nature and origin of graphed
structures.
INTERVIEWER: Can you find
one for me where the, the right
coordinate is bigger than the left
coordinate?
MARJORIE: If I go to the right
of the -and stay in the
positive.{cursor at (21,0)}
A-priori codes
Code Description Example
Math-1: Disequilibrium
Quotes showing moments where
the subject is "off-balance"
mathematically - "Disequilibrium
occurs when learners are presented
with new information they must
accommodate" (from my
comments on Vander Zander
1989).
?Okay I?m confused now again?
Math-2: Equilibrium
Quotes where the subject appears
to reach a point of understanding
about something he or she found
confusing
?[T]he reason why I can?t do it
that way is because?
Math-3: Functional thinking
Coded where the subject directly
connects the input with the output
value of a function
?Because no matter what it?s a
multiply of three and you?re
always going to add one to it.?
268
Code Description Example
Math-4: Mathematical
misconceptions
Quotations which reveal the
subject?s mathematical
misunderstandings.
?and it has like f times x equals
A {reading f(x) = A and cursor
pointing along f(x)}?
Math-5: Observing patterns
Quotes which show the subject
observing patterns.
?the pattern will continue with .
. . two squares . . . two
rectangles, two circles, and a
rectangle.?
Math-6: Problem Solving,
reasoning, and sense-making
Quotes where the subject appears
to be building knowledge by
making conjectures, drawing
logical conclusions, or connecting
new topics with existing
knowledge.
?INTERVIEWER . . . where
do you think number 11 would
go, step 11. Where would that
point fall? If we graphed that -
can you predict
MARJORIE: Add 3 to that,
should go to 34?
Math-7: Recursive thinking
Coded where the subject uses the
rate of increase or decrease to
describe a function
?I notice that each pattern
is different between each
one increasing at the base
and also right down the
middle and in between the
middle it just changed each
time.?
Code Family: Representational Ideas and Issues
Emergent codes
Code Description Example
Rep-IM: Indicative
movements
Gestures and mouse movements
made by the subject as they
interact with mathematical
representations
?{gesturing from step one to step
two}?
Also:
?{cursor waves back and forth
between the x and y
coordinates}?
Rep-IU: Idiosyncratic use
of representations
Quotes in which the subject used
standard representations in an
idiosyncratic way which reflects
the subject's mathematical
thinking
?INTERVIEWER: How do you
know what the - if we?re just
looking at that place where the
arrow is pointing right now, that
point right there, what would the
table . . .
MARLON: Oh, that would be
zero, that would be zero, nine, so
that would be zero (14:49). +9
{he writes 0+9}?
269
Code Family: Representational Ideas and Issues (cont.)
A-priori codes
Code Description Example
Rep-1: Endurance
Quotes showing places where a
topic either did or did not endure
for the subject from one session
to the next or through the course
of a session
?I can?t remember what they
actually do?
Rep-2: Internal
representations
Quotes which seem to give
possible insight into the subject's
internal representations. Some
ways this occurs are when the
subject discusses the external
representation so that we see how
he or she uses that representation,
or how he or she thinks about it.
?because I only have one point on
there, so it?s only going to give
me . . . one measurement.?
?if I go over here [31:38] {Moves
B to about (-4.5, 6.1)} I?m in a
negative area here still in the
positive area going upwards?
Rep-3: Multiple
representations
Quotes in which the subject is
working with different forms of
representation of the same
mathematical concept. These may
be:
? Pictures
? Numbers
? Graphs
? Geometric figures
? Tables
? Algebraic Expressions
or Equations
? Coordinate pairs
INTERVIEWER: . . . Where
would that point fall? If we
graphed that ? can you predict . .
.
MARJORIE: Add 3 to that,
should go to 34 . . . . Plot and hit
11, and 34. Yep . . . {she enters
11, 34 in the table}
Rep-4: Usefulness
Quotes indicating whether or not
a subject finds a particular
representation to be useful.
?INTERVIEWER: How would
you write it in a table? Write it as
if it were in a table.
MARLON: In a table so um it?d
be here , zero , plus 9 {he creates
an x, y table and enters 0 in the x
column and +9 in the y column}
That?d be my table?
Rep-5: Validity
Quotes which seem to show
whether a representation is valid
or not valid for the subject, that is
whether or not it accurately
represents the mathematics it
seeks to represent and is flexible
enough to allow additional
mathematical ideas to be built
upon it.
Because I think there was a
difference of three. For each um.
For at least the first three {points
to patterns on the paper}, there
was a difference of three. So I
had just added three and just kept
adding three. {points to pattern
number 4 and beyond}
270
Code Family: Representational Ideas and Issues (cont.)
Code Description Example
Rep-6: Mathematical
language
Quotes which give insight into
the subject's use of verbal
representations, such as his or her
misuse or idiosyncratic use of
terminology, difficulty in
remembering mathematical
terminology which affects his or
her ability to use the software, or
difficulty understanding what is
being asked of him or her
verbally by the interviewer.
Idiosyncratic use would be non-
standard, but make some sense to
the subject (such as "axle" instead
of "axis").
?I have the x and the new
parameter and it looks like a pi
sign and an e. So I click on new
parameter .See what that does.
Some coordinates are in there
{she is referring to the name and
value of the parameter as
?coordinates?}?
Rep-7: Visual observation
Where the subject seems to see
something in the way things look
that affects his or her thinking.
Among things to look for is the
phrase "I noticed"
?they?re all shaded?
271
Code Family: Influences and Uses of Technology
Emergent codes
Code Description Example
Tech-EM: Empowerment
Through Technology
Quotes showing the subject
making their own mathematical
choices and/or discoveries using
the technology
?Pretty neat. Let?s see what
measure is.?
Tech-WT: Technology as
a window into student
thinking
13
Quotes in which the subject's use
of technology gives the observer
insight into their mathematical
thinking
?now on the right hand side of
that is all positive {illustrates this
by moving the cursor from the
origin along the x-axis to the
right}?
Tech-SR: Technology as
an aid in the use of
standard representations
Quotes which seem to show that
the use of technology has aided
the subject in the use of a
standard representation with
which they may have been
struggling
?Where would that point fall? If
we graphed that - can you predict
MARJORIE: Add 3 to that,
should go to 34 . . . Plot and hit
11, and 34.Yep.?
Code Description Example
Tech-AR: Technology as
an aid to reasoning
Quotes which show the subject
using technology to reason about
mathematics
?INTERVIEWER: Yeah. Did it
do what you . . . thought it was
going to do?
MARLON: It?s going to - it?s
actually 5, 14 now where it
stopped it at . . . That would be
um, {writes g(4) = 4 + 9 and then
g(5) = 5 + 9}. Okay in this case
here . . . 5 plus 9 is 14
Tech-MC: Technology as
an aid to mathematical
communication
Quotes in which the subject uses
or expresses a desire to use the
software to communicate
particular mathematical ideas.
Okay same thing (here) if I go
over here [31:38] {Moves B to
about (-4.5, 6.1)} I?m in a
negative area here still in the
positive area going upwards. So
this x and y, okay. Same thing
down here {moves B to 3
rd
quadrant}.
Tech-CM: Using
technology to reveal and
clear up misconceptions
Quotes in which the use of
technology helps identify and/or
clear up a misconception
?and hit okay . . . Oop. Okay I
did it wrong?
13
Stevens, To, Harris, and Dwyer (2008) spoke of LOGO as giving a ?window into the student?s
mind? (p. 199).
272
Code Family: Other
These codes are not part of the final theory, but were used to note characteristics of data
and were provided so that inter-coder reliability tests could be accurately conducted.
Code Description Example
Other-1: Characteristics
of the learner
Quotations which give
information about the background
or attitude of the subject.
?I used to be in the military?
Other-2: Clerical and
technical procedures of
the Study
Portions where papers are being
signed, the use of cameras and
other arrangements are being
described, and the methodology
of the study is being explained.
?move this motion controller up,
because when I put the films
together your paper will be down
there.?
Other-3: Incidental
conversation
Casual conversation such as
discussions about the weather, the
temperature of the room,
greetings, good-byes.
Conversation not related to the
topic of the study or how it is
being conducted.
?I?m sorry it was so hot in here?
Other-4: Narrative notes
about the study
Not part of the transcription of
what happened during the study ?
environmental or background
information added in later.
Marjorie did not create or interact
with any mathematical
representations on paper during
this session.
Other-5: Scaffolding
Places where particular
statements or actions on the part
of the interviewer are influencing
the actions of the subject.
?And what else do you see
happening in that picture??
273
Example of coding
Unit Codes Reasoning
INTERVIEWER: Okay, so you don?t
have any existing values in your sketch.
So you can click on those menus and
see what you find.
MARJORIE: Okay, click on values
{she does}
INTERVIEWER: Go ahead (values)
MARJORIE: I have the x and the new
parameter and it looks like a pi sign and
an e {stated as cursor moves down the
list of these items}. So I click on new
parameter. See what that does. Some
coordinates are in there {cursor moves
along the name and value of the
parameter as she says "coordinates"}
um, under the name, it looks like they
have a bracket and one and then under
it says equal value one point
Indicative movements
Mathematical misconception
Scaffolding
Verbal/mathematical
vocabulary/language
The cursor
movements
which are
present
She confuses
the name and
value of a
parameter with
?coordinates?
The
interviewer
suggests
actions on the
part of the
subject
Misuse of the
word
?coordinates?
274
Appendix E
Consent Forms
275
IRB approval notice
276
277
278
RECRUITMENT SCRIPT (verbal, in person)
(This should be a brief version of the consent document.)
My name is Lauretta Garrett, a doctoral student from the Department of Curriculum and
Teaching at . I would like to invite you to participate in my research
study to find out how using computer software to study mathematics affects how a person
thinks about mathematics. Anyone may participate if they are enrolled in Math 098,
Math 099, or Math 100 and age 18 or older. Please do not participate if you are under
age 18.
As a participant, you will be asked to participate in a sequence of video taped interviews.
Your total time commitment after the initial interview will be approximately 30 to 45
minutes per interview, for four to eight interviews over a period of about 4 to 8 weeks.
(Briefly discuss any risks, compensation or benefits, costs, privacy issues, or other
information that would likely influence the participant?s interest in the study)
By participating, you risk the possibility of your personal information being known to
others, but I will take steps to protect that data. You also risk some discomfort if you
suffer from mathematics anxiety, but you will be able to work at your own pace. You do
have the opportunity to learn about mathematics software and mathematics topics from
your course. Your personal information will be kept secure during the study and will be
destroyed when the research is completed You can withdraw at any time. You can decide
who, in addition to me and possibly one other person involved with the research, will see
the videotapes.
If you would like to participate in this research study, contact me at the email address or
phone number listed on this flyer (pass out flyers).
Do you have any questions now?
If you have questions later, my contact information is on the flyer.
279
Technology Study
HAVE FUN! LEARN! PREPARE
FOR THE FUTURE!
Are you enrolled in Math 98, Math 99, or Math 100 and age 18 or
older?
Do you want to learn about some mathematics software that might help
you to be successful in mathematics?
If you answered YES to these questions, you may be eligible to participate in
a mathematics education research study.
The purpose of this research study is to find out how using computer
software affects the way people think about mathematics. Benefits include
learning about a type of technology used for mathematics education and
receiving personal instruction in a topic that is part of the curriculum for the
course in which you are currently enrolled. Participants will receive gift
certificates in appreciation for their participation.
Anyone 18 or older enrolled in Math 97, Math 98, or Math 100 is eligible.
This study is being conducted by a doctoral student in the Department of
Curriculum and Teaching at . Interviews will take place at
.
Please contact Lauretta Garrett at or for
more information.
PLEASE RESPOND BY SEPTEMBER 25th