The availability of internet library resources, including UMI’s archive
of digital dissertations (http://wwwlib.umi.com/dissertations), has the
potential to link studies and enhance what we are learning from
research on undergraduate mathematics education. It is now
possible to read full text dissertations that have been selected based
on keywords found in the dissertation’s abstract. A 6/30/06
search using the keyword “mathematics education”, for example, yields a
total of 7,598 dissertation abstracts; 342 of these represent degrees
awarded in 2004. Of these 342 dissertations, 115 were awarded by
the 46 institutions posted (6/29/06) on the SIGMAARUME web as
institutions offering doctoral programs in mathematics
education. If separated from the research conducted in k12
settings, a partial view of the dissertation research focusing on
undergraduate mathematics (n=29) completed in one year emerges.
This preliminary report and discussion about dissertation research has
the potential to help understand and guide future research, practice,
and policy development.
Assessments
that Improve Proof Writing Skills
Casey Dalton
University of Northern Colorado
daltonw13@yahoo.com 
Nissa
Yestness
University of Northern Colorado
nissa.yestness@unco.edu 
Hortensia
SotoJohnson
University of Northern Colorado
hortensia.soto@unco.edu 
In this qualitative study, we investigate what and how assessments help
students improve their proof writing skills. To investigate these
questions, students who completed 2nd semester of an undergraduate
abstract algebra course were interviewed and also completed an
openended questionnaire. Data from the interviews were
transcribed and coded using grounded theory methodologies as described
by Strauss and Corbin (1998). The three themes that emerged as
contributors to the improvement of students’ proof writing skills are
practicing writing proofs, observing proofs being done by others, and
receiving feedback on proofs. The assessments with which these
themes were reported were identified. Students reported that
inclass proof presentations provided an opportunity to engage with all
three themes, and homework provided opportunities to practice writing
proofs and receive feedback on proofs. These results indicate
that assessments that involve these three themes should be used in a
classroom where improving students’ proof writing skills is an
objective.
Curtis
Bennett
Loyola Marymount University
cbennett@lmu.edu

Jacqueline
Dewar
Loyola Marymount University
jdewar@lmu.edu 
This article presents the results of a study of the evolution of
students’ understanding of proof as they progress through the
mathematics major curriculum at a mediumsized comprehensive
university. The study attempted to identify which courses or learning
experiences promote growth in student understanding of proof. The
researchers employed surveys and thinkalouds on proof. As an aid in
analyzing thirteen thinkalouds on proof (12 students, 1 faculty), the
researchers developed a typology of mathematical knowledge that
includes six cognitive and two affective components. Then the typology
was expanded into a taxonomy that describes the journey toward
proficiency in each of these components. The resulting mathematical
knowledgeexpertise taxonomy can be applied to analyze student work and
instructor responses.
2007 BEST PAPER AWARD, Honorable Mention
The research reported here investigates the question, what is the
nature of students’ understanding of firstorder differential equations
[FODEs] in a modern course on ordinary differential equations [ODEs]?
Modern courses on ODEs emphasize analytical, numerical, and qualitative
solution methods and hence use graphical and algebraic representations
of ODEs and their solution functions. Sfard’s (1991, 1994) theory of
reification predicts that a deep understanding of function is a
necessary component upon which understanding of FODEs is built; in the
terminology of the theory one must have reified the concept of
function. The two case studies reported on in this talk show a contrast
in understanding that gives insight into the cognitive importance of a
reified notion of function to the development of FODE understanding in
a modern, multirepresentational approach to their study.
Nicole Engelke
California State University, Fullerton
nengelke@fullerton.edu 
Related rates problems are a source of difficulty for many calculus
students. There has been little research on the role of the mental
model when solving these problems. Three mathematicians were observed
solving three related rates problems. From the examination of their
solutions, a framework for the solution process emerged. The framework
is based on five phases: draw a diagram, construct a functional
relationship, relate the rates, solve for the unknown rate, and check
the answer for reasonability. Each phase can be described by the
content knowledge the problem solver accesses, the mental model that is
developed, and the solution artifacts that are generated.
Susan S.
Gray
University of New England
Biddeford, Maine
sgray@une.edu 
Barbara J.
Loud
Regis College
Weston, Massachusetts
barbara.loud@regiscollege.edu 
Carole P.
Sokolowski
Merrimack College
No. Andover, Massachusetts
carole.sokolowski@merrimack.edu 
The study of calculus requires an ability to understand algebraic
variables as generalized numbers and as functionallyrelated
quantities. These more advanced uses of variables are indicative of
algebraic thinking as opposed to arithmetic thinking. This study
reports on entering Calculus I students’ responses to a selection of
test questions that required the use of variables in these advanced
ways. On average, students’ success rates on these questions were less
than 50%. An analysis of errors revealed students’ tendencies toward
arithmetic thinking when they attempted to answer questions that
required an ability to think of variables as changing quantities, a
characteristic of algebraic thinking. The results also show that
students who more successfully demonstrated the use of variables as
varying quantities were more likely to earn higher grades in Calculus I.
Jon
Hasenbank
University of Wisconsin  La Crosse
hasenban.jon@uwlax.edu

Ted Hodgson
Montana State University
hodgson@montana.edu 
This study examined the effectiveness of instruction based upon Burke’s
(2001) Framework for Procedural Understanding. The Framework is
designed to help students develop deep procedural knowledge, which
presumably facilitates recall and promotes future learning. The
quasiexperimental design paired six college algebra instructors
according to teaching experience, and the instructional treatment was
assigned to one member of each pair. Students’ ACT/SAT scores
established the equivalence of treatment and control groups. Data
consisted of classroom observations, homework samples, common hour
exams, procedural understanding assessments, supplemental course
evaluations, and interviews with treatment instructors. An ANCOVA
revealed that treatment group students scored significantly higher than
control group students on procedural understanding. Moreover, although
treatment students were assigned fewer drill questions, no significant
differences were detected in procedural skill. Overall, students
possessing procedural understanding exhibited greater procedural skill,
regardless of instructional approach. Interviews with treatment
instructors revealed implementation issues surrounding Frameworkbased
instruction.
BEST PAPER AWARD, 2007 CONFERENCE 

David Kung
St. Mary's College of Maryland
dtkung@smcm.edu 
Natasha
Speer
Michigan State University
nmspeer@msu.edu 
Just as doing mathematics creates opportunities to learn mathematics,
“doing teaching” creates opportunities to learn to teach. Nowhere is
this more applicable than for graduate students who have little or no
teaching training prior to their first teaching assignments. We report
on our analysis of how the research literature on teachers’ onthejob
learning can be applied to the context of graduate student professional
development. We combine this analysis with our synthesis of findings
about the role of teachers’ knowledge about student thinking in shaping
instructional practices and student learning opportunities. Our
findings take the form of a framework, grounded in research on teacher
learning, to guide the design of activities andprograms to equip
graduate students with the skills and dispositions to inquire into and
learn from their teaching experiences.

Elise
Lockwood
Portland State University
elockwoo@pdx.edu 
Craig
Swinyard
Portland State University
swinyard@pdx.edu

The purpose of the ongoing research is to contribute to the development
of a conceptual analysis of the formal definition of limit. The
research is developmental in nature, consisting of a threestep
iterative cycle designed to accomplish two purposes: 1) to provide the
participants with optimal opportunity to come to reason coherently
about the formal definition of limit and, in so doing, 2) to produce
empirical evidence that will enable the identification of what we term
the ‘conceptual entailments’ of students’ reasoning about the formal
definition of limit.
Ami Mamolo
Simon Fraser University
amamolo@sfu.ca 
This study explores views of infinity of firstyear university students
enrolled in a mathematics foundation course, prior to and throughout
instruction on the mathematical theory involved. A series of
questionnaires that focus on geometrical representations of infinity
was administered over the course of several weeks. Along with
investigating students’ naïve conceptions of infinity, this
enquiry also examines changes of those views as beliefs, intuition, and
instruction are combined. The findings reveal that students’
conceptions about the nature of points, for instance, prevented them
from drawing any correlation between numbers and points on a number
line. Furthermore, a preliminary theoretical analysis using an
APOS framework asserts that participants conceive of infinity mainly as
a process, that is, as a potential to, say, create as many points as
desired on a line segment to account for their infinite number.
Bernadette Mullins
BirminghamSouthern College
bmullins@bsc.edu 
This presentation describes research undertaken by a partnership
including nine school districts, two institutes of higher education,
and a nonprofit organization. The partnership has made major
revisions to course offerings and support systems for preservice and
inservice mathematics teachers. After a review of the literature and
of existing undergraduate programs, and discussion between district and
IHE partners, a new track of the mathematics major designed
specifically for future middle school mathematics teachers was
developed at one of the institutes of higher education. The partnership
also offers seven mathematics content courses during the summer
available to both inservice teachers and preservice teachers (these
may be taken for university credit or professional development
hours). Preliminary results in this presentation describe changes
in the mathematics content knowledge of the preservice and inservice
teachers, the classroom practice of inservice teachers, and the
mathematics content knowledge of the middle school students.
Recently mathematics educational researchers have taken an increasing
concern in the teacher¹s discourse move, which is defined as a
deliberate action taken by a teacher to participate in or influence the
discourse in the mathematics classroom (Krussel, Edwards, &
Springer, 2004). This study explored the roles of revoicing in the
undergraduate inquiryoriented mathematics class in the perspective of
teacher¹s discourse move. The data for this analysis came from
four classes about phase portrait of the system of differential
equations with initial value from a large state university. We
particularly analyzed revoicing linked with questioning, telling, and
directing through the result of coding of teacher¹s discourse
move. The results show that revoicing has the following roles: bonding
students¹ response to the teacher¹s discourse move 
questioning, telling, or directing; providing students the ownership of
knowledge; providing students the springboard for further thinking.
As part of ongoing research into cognitive processes and student
thought, we investigate the interplay between mathematics and physics
resources in intermediate mechanics students. In the mechanics
course, the selection and application of coordinate systems is a
consistent thread. Students start the course with a strong
preference to use Cartesian coordinates. In small group interviews and
in homework help sessions, we ask students to define a coordinate
system and set up the equations of motion for a simple pendulum where
polar coordinates are more appropriate. Using a combination of
Process/Object Theory[1] and Resource Theory[2], we analyze the video
data from these encounters. We find that students sometimes
persist in using an inappropriate Cartesian system. Furthermore,
students often derive (rather than recall) the details of the polar
coordinate system, indicating that their knowledge is far from solid.
We present preliminary findings of a case study through which we
sought, through detailed analysis of four students’ arguments, to
distil a set of analytic constructs that might help make clearer sense,
in general, of how arguments conveyed through special cases might
support assertions that are understood to hold in general. We began
analysis from a particular standpoint: to focus fundamentally on the
learners’ representations and on how they reasoned from them. We found
it helpful to distinguish two perspectives to guide the subsequent
analysis. On the one hand, we direct detailed attention to how learners
reason, most especially on how they organize the logic of their
arguments. On the other hand we seek to understand the learners’
representations through the way they structured them, and through how
such structures might be reshaped or reframed over time.
Many beginning university students struggle with the new approaches to
mathematics that they find in their courses due to a shift in
presentation of mathematical ideas, from a procedural approach to
concept definitions and deductive derivations, with ideas building upon
each other in quick succession. This paper highlights this situation by
considering some conceptual processes and difficulties students find in
learning about eigenvalues and eigenvectors. We use the theoretical
framework of Tall’s three worlds of mathematics, along with
perspectives from processobject and representational theory. The
results of the study describe the thinking about these concepts of
groups by first and second year university students, and in particular
the obstacles they faced, and the emerging links some were constructing
between parts of their concept images formed from the embodied,
symbolic and formal worlds. We also identify some fundamental problems
with student understanding of the definition of eigenvectors that lead
to problems using it, and some of the concepts underlying the
difficulties.
This presentation will relate to a range of
contemporary
theoretical perspectives by presenting a framework of three modes of
thinking
that operate so differently as to present essentially three distinct
‘worlds of
mathematics’—
conceptual embodiment,
proceptual symbolism and axiomatic
formalism—which will be abbreviated in the context of this
theoretical
framework to
embodiment,
symbolism and
formalism. Human learning will be
addressed in terms of compression of knowledge in which important
aspects of a
situation are named and built into rich
thinkable concepts that enable
thinking
to be performed by making
connections
between the thinkable concepts to build
successively more sophisticated conceptual structures. In this way, any
new
concept needs to be seen in the light of previous experience, building
on
aspects within the individual’s concept image that I call
metbefores.
Symbolism builds on earlier embodied and symbolic metbefores;
formalism builds
on earlier embodied, symbolic and formal metbefores. All three operate
with
ongoing interchange between them.
To illustrate the power of embodiment to
underpin formal
thinking, the embodied notion of local straightness will be contrasted
with the
symbolic notion of local linearity to show that while the latter has
great
computational power, it lacks the embodied meanings that lead naturally
to
significant formal meanings in mathematical analysis. On the other
hand,
proceptual computations in many areas such as groups, vector spaces,
mathematical analysis will be shown to lead not only to formal
definitions but
also to structure theorems that provide deeply meaningful embodiments.
The presentation will consider a range
of
recent studies
which show how embodied metbefores can both enhance and impede formal
thinking
and discuss how a combination of (conceptual) embodiment, (proceptual)
symbolism and (axiomatic) formalism relates to the allembracing
Lakoffian
concept of embodiment, and processobject encapsulation that is the
main focus
of APOS theory.
Pushing more radically Duval's (1991) research orientations, and taking
into account reflexions initiated in Author (2005) regarding the
obstacle we identify as ‘hindering truth value’, we designed tasks in
which pupils fit into an oriented graph the given propositions of a
geometrical proof. An experiment involving a sequence of three such
tasks was conducted in the spring of 2005 and of 2006. Among the
conclusions drawn from the data that were gathered: the role played by
rules of inference is underestimated by pupils; teams that took the
rules into account at each step were better achievers than teams that
(tried to) set the rules afterwards; teams that worked from the end
forward to the beginning were also better and faster achievers; the
organizational work required by the proposed tasks contributed to
fostering pupils' understanding of the mechanisms that rule the
deductive structure.
Affecting
Secondary Mathematics Teachers’ Instructional Practices by Affecting
their Mathematical Knowledge
Pat
Thompson
Arizona State University
pat.thompson@asu.edu 
Marilyn
Carlson
Arizona State University
marilyn.carlson@asu.edu 
Irene Bloom
Arizona State University
irene.bloom@asu.edu 
Sharon Lima
Arizona State University
lamamushka@gmail.com 
Chris Miller
Arizona State University
christina.miller@asu.edu 
Jeff Rodel
Arizona State University
jeffrey.rodel@asu.edu 
Carlos
CastilloGarsow
Arizona State University
cwcg@asu.edu 
Ana E. Lage
Arizona State University
Ana.Lageramirez@asu.edu

Seven papers, spread over two sessions, will report on the first 18
months of a fiveyear project that aims to improve secondary
mathematics teachers’ instruction while at the same time producing
theoretical understandings of the intervention’s effects on teachers
beliefs, values, and knowledge. One paper will give an overview of the
project and a broad summary of its results. Five other papers will
report on the effects of covariational approaches to functions on
teachers’ reasoning, on making meaning as a means for professional
development, on decentering as a critical component of teacher
practice, on teaching for meaning in trigonometry, and on an extension
of the project into one classroom.
Proving
Styles in Advanced Mathematics
Keith Weber
Rutgers University
khweber@rci.rutgers.edu 
Lara Alcock
University of Essex
lalcock@essex.ac.uk

Iuliana Radu
Rutgers Unviersity
tenis@rci.rutgers.edu 
Research in advanced mathematical thinking suggests that there are at
least two qualitatively distinct ways that students may productively
reason about advanced mathematical concepts (e.g., Vinner, 1991; Raman,
2003). Individuals can reason about concepts by focusing primarily on
their definitions and using logic to deduce properties about the
concept from these definitions. Alternatively, they coordinate their
image of the concept (cf., Tall & Vinner, 1981) with its formal
definition and use both to determine what properties the concept may
have (e.g., Vinner, 1991; Pinto & Tall, 1999). Although both
modes of reasoning are worthwhile, several studies on students'
reasoning in real analysis suggest many students predominantly use only
a single mode of reasoning to think about formal concepts (e.g., Alcock
& Simpson, 2004, 2005; Pinto & Tall, 1999, 2002). The goal of
this presentation is to extend and generalize this research by
examining students' reasoning styles in another context and another
domain. Specifically, we will examine the ways that undergraduates
attempt to construct proofs in a transitiontoproof course. The goal
of this presentation is to document the existence of undergraduates'
proving stylesthat is, we will show that some students consistently
base their proof attempts on their informal images of the involved
concepts while other students never use their concept image when they
construct proofs and instead focus on logical rules and manipulations.
Ian Whitacre
San Diego State University
ianwhitacre@yahoo.com

This paper describes a classroom teaching experiment around number
sensible mental math in a course for preservice elementary teachers.
Number sense is a widely accepted goal of mathematics instruction, and
mental math is a hallmark of number sense. In order to foster its
development in their students, elementary teachers must have good
number sense themselves. The author designed an instructional sequence
aimed at students’ development of number sense through authentic mental
math activity. The theoretical orientation for this study can be
characterized as emergent. Students’ individual mathematical activity
is recognized as taking place in a social context, while the social
environment of the classroom is made up of individuals who contribute
to that community. Analysis of data suggests that students did develop
greater number sense as a result of their participation in classroom
activities. Particular instructional innovations represent significant
results that may be applicable to mathematics teaching at various
levels.
Sense of
Community: Contributing Factors and Benefits
Nissa
Yestness
University of Northern Colorado
Nissa.Yestness@unco.edu 
Hortensia
SotoJohnson
University of Northern Colorado
Hortensia.Soto@unco.edu 
Casey Dalton
University of Northern Colorado
Casey.Dalton@unco.edu

In this qualitative study we explore how assessments contribute to
building a sense of community (SOC) in the classroom of an
undergraduate abstract algebra course. Strike (2004) describes
community as a process rather than a feeling and outlines four
characteristics of community: coherence, cohesion, care, and contact.
Coherence refers to a shared vision; cohesion is the sense of community
that results from the shared vision; care is a necessity to initiate
one into the vision, and contact refers to structural features of the
community. Using a grounded theory approach we analyzed student
interviews and report on the contributing factors to SOC as described
by students as well as perceived benefits by these students. We found
that contributing factors to the SOC align with Strike’s 4 C’s
definition of community and fall into two large categories: teacher and
environment. The contributing factors provide a model for a teacher
that wishes to build a SOC in his classroom, and the benefits provide
support for doing so.
Short Papers
The researcher’s interest in finding ways to improve college algebra
has led to a study of learning styles of College Algebra students.
Since 77% of the surveyed college algebra students are more active
learners than reflective learners, a new approach is needed. A College
Algebra course was paired with Modern Dance to enable students to learn
both the dance and the math curricula, by using action to help
understand the mathematics. This presentation will focus on the ways of
teaching Transformations of Function in a kinesthetic way to the
students in the class. The presentation will discuss both the learning
that occurred and the student’s attitude toward the subject..
This study focuses on how students incorporate mathematical reasoning
into investigations using Dynamic Geometry Software (DGS), with a
special emphasis on how mathematical definitions are incorporated into
these investigations. Six upperlevel undergraduate students from
a comprehensive midwestern university in the United States of America
were asked to use DGS to justify three geometrical assertions in
individual, semistructured interviews. The students generally
incorporated correct definitions into their DGS investigations but had
difficulty parsing the mathematical statements and exhibited
difficulties similar to the ones experienced when using definitions in
proofs.
The use of complex numbers occurs throughout mathematics, engineering,
and science and undergraduates learn to use complex numbers in a
variety of courses, including calculus, differential equations, and
more advanced courses in complex analysis. Yet our review of the
literature on the teaching and learning of complex numbers at all grade
levels has, to date, revealed no empirical studies focused on this
important mathematical terrain. The proposed presentation will report
on one of the first empirical studies on student learning of complex
number, conducted during the last three weeks of the Fall 2006 semester
in a capstone mathematics course for prospective secondary school
mathematics teachers at a large southwestern university.
Axelle P.
Faughn
California State University Bakersfield
afaughn@csub.edu 
Terran
Felter Murphy
California State University Bakersfield
tfelter@csub.edu 
Interviews of students entering college trigonometry reveal high levels
of math anxiety; having to remember numerical tables and an
overwhelming amount of formulas unrelated to meaningful representations
are to be blamed for such apprehension. Furthermore students often
perceive trigonometry as disconnected from other mathematical topics
with no transition from previously acquired knowledge. Technology has
proven efficient in enhancing students’ learning by allowing them to
rediscover mathematical properties through visual manipulations. In
particular research on using Graphing Calculators, Excel spreadsheets
or The Geometer’s Sketchpad in middle school and high school
trigonometry courses show positive results for achieving higher
conceptual understanding. In this project the researchers propose to
make effective use of a unique lab setting in order to investigate the
effect of the interactive geometry software, The Geometer’s Sketchpad,
in a college trigonometry course. We will present activities designed
to support meaningful explorations of trigonometric concepts from a
constructivist perspective.