Online Proceedings
for the Thirteenth SIGMAA
on Research in Undergraduate Mathematics Education Conference
Marriott Raleigh City
Center  Raleigh, North Carolina
February 25 â€“ February 28, 2010
Listed alphabetically by author. Links to PDFs appear beneath title
when available.
Student
Understanding of Partial Derivatives in Physical Chemistry
Nicole Becker Purdue
University 
Renee Cole
University of Central Missouri 
Chris
Rasmussen San Diego
State University 
Marcy Towns Purdue
University 
Upperlevel undergraduate physical chemistry courses require students to be proficient in calculus in order to develop an understanding of thermodynamics concepts. Here we will present the findings of a study that examines the relationship between math and chemistry in two undergraduate physical chemistry courses. Students participated in thinkaloud interviews in which they responded to a set of questions involving mixed second partial derivatives with either abstract symbols or thermodynamic variables. Preliminary results and analysis of the study will be discussed.
Jason K. Belnap The University of Wisconsinâ€”Oshkosh 
Discussions are present in and central to most
instructional forms. It is of upmost importance to understand the nature of
student involvement in classroom discourse, particularly in learnercentered
classrooms. The nature of student involvement and quality of instruction may be
revealed through analysis of classroom discussion; however, student involvement
complicates discussion structure and discourse analysis.
While studying unfacilitated discussions among
teachers, in a professional development program, Belnap and Withers (2008)
developed a framework describing how individual contributions construct a
discussionŐs content. I plan to extend this framework to the classroom context,
in order to reveal the nature of student involvement in classroom discourse.
This presentationŐs discussion will focus on
preparing for such a study. Questions include: What critical differences are
there between professional development and classroom contexts? How should these
be addressed in study design? and What literature may
be related to, add to, or inform this effort?
Irene Biza University of East Anglia (UK) 
This study focuses on a teaching experiment applied in a Year 12 class
for the introduction of the derivative and the tangent line of function graph.
This experiment was based on research results concerning studentŐs perceptions
about tangents when they had met the notion of tangent in different
mathematical contexts (Geometry and Analysis). For the experimental needs an
electronic environment was developed, utilizing Dynamic Geometry software. The
analysis focuses on the evolution of classroom mathematical discourse with the
mediation of the electronic environment and with specific examples. Here I
focus on an incident of the experiment. This incident exemplifies: how an image
of a curve magnified in order to look straight in the electronic environment
did not act as visual mediator for a
student towards a claim that a curve has a tangent; the conflict in
mathematical discourse about tangents; and, its resolution through the
discussion of a particular example.
Testing
Conceptual Frameworks of Limit: A ClassroomBased Case Study
Timothy Boester Wright State University 
The
purpose of this study is to test three proposed conceptual frameworks of how
students come to understand informal (dynamic) and formal (static) definitions
of limit: one based on embodied cognition (Lakoff & Nunez, 2001), one based
on APOS theory (Cottrill et al., 1996), and one created by the researcher.
Predictions are made of how students would respond, based on each conceptual
framework, to instructional tasks posed in a firstsemester college calculus
discussion section. These predictions are then compared to the responses of
eight students during a sequence of interviews spanning the course of the
semester. The results support the conceptual framework proposed by the
researcher, with specific responses suggesting a further refinement, explaining
how students come to bridge dynamic and static conceptions of limit.
Assessing
Proofs with Rubrics: The RVF Method
David E. Brown and Shayla
Michel Utah State University Department of Math and
Statistics 
We present an easytoimplement 3axis rubric
for the formative and summative assessment of openended solutions and proofs.
The rubric was constructed for the use on the written work of students in a
Discrete Mathematics class at a researchoriented university, with the
following in mind: (1) To aid in the efficiency and consistency of assessment
of proofs and openended solutions, with the possibility of being comfortably
implemented by an undergraduate assistant; (2) To provide the simultaneous
formative and summative assessment of the studentsŐ written work. Consequently,
the questions we addressed for the rubricŐs construction are: (1) How can we foster good technical writing skills in a way
that improvement can be measured? (2) How can large amounts of written work be
processed and assessed so that summative and formative judgments are passed but
without much time used by the instructor/professor/TA? The rubric we devised
operates with the categories readability, validity, and fluency, (whence,
ŇRVFÓ) corresponding to (respectively) the ease with which the solution or
proof can be read, correctness of calculations and deductions, and the extent
to which a student is able to use and communicate via the technical notions
relevant to the problem or proof. In order to show the independence of the
categories, we will give examples of solutions with high scores in any one or
two of the categories only. The rubric format is communicated to the students
and discussed in class before any written work is assessed. The rubric has been
implemented by professors and teaching assistants only after being trained in
its use.
The
Role of Skepticism and Uncertainty in the Emergence of the Practice of Proving
Stacy A. Brown Pitzer College 
The
purpose of this paper is to contrast uncertainty and skepticism in terms of
their role in the emergence of the practice of proving. In particular, drawing
on data from a series of teaching experiments, distinct paths to a disposition
of skepticism are illustrated. These paths to skepticism are contrasted with
the paths to uncertainty identified by Zaslavsky (2005). This analysis points
to the dynamic interplay between the unknown and belief, which is foundational
to the emergence of the practice of proving in classroom communities.
Inclusion
of Students with Disabilities
Joanne C. Caniglia Kent State University 
The
purposes of this preliminary study are to explore the beliefs of future mathematics and special education
teachers regarding inclusionary practices and to investigate reasons for differences if they exist.
Twentytwo secondary mathematics
preservice teachers and 17 special education teachers participated in the study. Q methodology was used to analyze 25 statements regarding
the mathematics teacherŐs role, attitude and
knowledge of collaboration and disabilities,
the role of special educators in inclusion, and the impact of special education students in inclusive
settings. Preliminary results
indicate that preservice teachers have a positive attitude toward
inclusion. Some differences, however, were found between the attitudes
of special and mathematics preservice teachers.
The results of this study may assist in the identification of areas of need for preservice as well as graduate
coursework pertaining to inclusive education in an era of No Child Left Behind and the Individuals with
Disabilities Education Act (IDEA).
Carlos CastilloGarsow Arizona State University cwcastil@asu.edu 
Derek,
an Algebra II student, was given the task of evaluating the same model that he
had used for continuously compounding financial growth in a context of human
population growth. Although Derek was tasked with using the same model, the conclusions that he reached about the financial account and the
population were influenced by his understanding of the problem context.
In the financial situation, Derek imagined that the account grew continuously
though every real number, while in the population context, Derek first
concluded that the same model grew discretely, and then later concluded that
the model predicted unrealistic continuous growth.
This
paper examines DerekŐs interaction with his idea of a mathematical model as he
engaged in these tasks. Specifically it argues that Derek never distinguished
between a ŇrealÓ world and a ŇmathematicalÓ world that was a model of the
ŇrealÓ worldâ but rather developed the situation and the mathematics together.
Renee Cole
University of Central Missouri 
Marcy Towns
Purdue University

Chris
Rasmussen San Diego
State University 
Nicole Becker Purdue
University 
George Sweeney San Diego
State University 
Megan Wawro San Diego
State University 
Physical chemistry is a subject that uses
mathematical inscriptions to carry chemical meaning. In order to gain
understanding, both curricular and pedagogical, of how students build an
understanding of mathematical inscriptions that are used in chemistry, it is
necessary to document student reasoning and classroom practices. A threephase
approach grounded in ToulminŐs argumentation scheme was developed to trace the
growth of ideas in an inquiry classroom. This method of
documenting collective production of meaning was adapted for use in analyzing
an inquiryoriented physical chemistry classroom. The
difference in classroom structure necessitated modifications to the application
of the methodology, but the analysis provided empirical evidence for common
themes that define classroom chemistry practice.
This evidence will be presented along with the implications for
instructional design and teaching.
Exploring
Teachers' Capacity to Reflect on their Practice
Scott Courtney Arizona State University 
Although
the idea of pedagogical content knowledge (PCK) has been elaborated in numerous
studies, there has been little clarification of what constitutes it or research
into its development. Furthermore, studies that have investigated PCK, or
mathematical knowledge for teaching (MKT) as first introduced by Thompson and
Thompson (1996), have historically focused on preservice teachers at the
elementary level. This study contributes to filling these voids by
investigating inservice secondary school teachersŐ ways of thinking that
supported or constrained their capacity to reflect on their practice as they
engaged in activities designed to promote powerful mathematical knowledge for
teaching as proposed by Silverman and Thompson (2008). Findings indicate that
teachers whose personal mathematics focused on facts and skills found
reflection most difficult; their mathematical knowledge constrained their
capacity to reflect on the reasoning that they engaged in through instruction,
impeding the level of coordination of meanings required to sustain propitious
reflection.
Communal
Communication in Undergraduate Real Analysis: the Case of Cyan
Dr. Paul Christian Dawkins The University of Texas at Arlington 
Vygotsky
(1978) and Cobb, Wood, & Yackel (1993) have especially informed the
research community about the role social interaction plays in individual
learning, however much of their work has been with younger children. Social and
interactive learning generally stands in contrast to the traditions of
proofbased classrooms (Weber, 2004), but this study describes the manner in
which one undergraduate real analysis student built an intellectual community
around himself to facilitate his learning and the learning of his classmates.
He pursued multiple pathways of communication and described some of the
benefits of each. I discuss the classroom norms and departmental culture that
facilitated this studentŐs social learning.
How
do Undergraduate Students Navigate their Example Spaces?
Anthony Edwards Loughborough University
(UK) 
Lara Alcock Loughborough University (UK) 
In this
presentation we will report on a study where first year undergraduates were
asked to generate examples of real sequences satisfying certain properties.
Following earlier work by Antonini (2006), who classified the example
generation strategies of expert mathematicians into three types—trial and error, transformation and analysis—we
use a graphical representation of Watson and MasonŐs (2005) construct of
example spaces to explicate AntoniniŐs classification, and extend it to include
cases of falsetransformation and falseanalysis.
Point/Counterpoint:
Should We Teach Calculus Using Infinitesimals?
Robert
Ely Department of Mathematics University of Idaho 
Timothy Boester Wright State University 
During the first 150 years of its life, calculus
was developed and widely applied by mathematicians who conceptualized and
notated integrals and derivatives using infinitesimals. Although the rigorous notion of limit
took the place of infinitesimals, it has been since shown that infinitesimals
can be used to define calculus with equal rigor. So why should calculus not be taught today using
infinitesimals? This paper
presents a point/counterpoint debate about the merits and drawbacks of the
infinitesimal approach to calculus, appealing to educational research findings,
issues of notational affordance, formal abstraction, and the various student
conceptions of limits as dynamic and static entities.
Counting Two Ways: The Art of Combinatorial
Proof
Nicole Engelke California State University Fullerton 
Todd CadwalladerOlsker California State University Fullerton 
Combinatorial proofs are used to show that many
interesting identities hold. Typically, after examining an identity, one poses
a counting question and proceeds to answer it in two different ways. This poses
a challenge for students as it requires a way of thinking other than they have
traditionally encountered. The newly introduced proof technique requires
students to either create new strategies or adapt their old strategies to write
such proofs. We will discuss the results of a preliminary study on the
combinatorial proofs written by students in an upperdivision combinatorics
course and a graduatelevel discrete mathematics course. In particular, we will
identify some common difficulties that students have and suggest ways to
overcome them.
Conceptualizing
Multivariable Limits: From Paths
to Neighborhoods
Brian Fisher Pepperdine University 
It
is well accepted that the limit concept plays a foundational role in
presentday calculus education. At
the same time, there is widespread agreement among both educators and
researchers that most students struggle to develop a solid understanding of
this important idea. In this preliminary report I will discuss the results of
two teaching experiments exploring the concept of limit in multivariable
calculus. I will describe how
students participating in these experiments changed the way they view the
concept of limit by changing their emphasis from dynamic motion placed upon
paths to an emphasis of closeness.
Modeling
Mathematical Behaviors; Making Sense of Traditional Teachers of Advanced Mathematics Courses Pedagogical Moves
Tim FukawaConnelly University of New Hampshire 
This study investigates proof writing strategy within a traditionally
taught abstract algebra classroom. Drawing on Rasmussen and MarrongelleŐs
(2006) construct of Pedagogical Content Tools (PCTs) I expand the domain of
analysis to include traditional instruction, and increase the number of PCTs
under consideration. I describe
how the instructor modeled behaviors that are important in learning advanced
mathematics and characterize this a broad category of PCTs called Modeling
Mathematical Behavior. Proofwriting was one of the
most important of the classroom activities that I observed. During proof discussions, the
instructor made significant use of questions, both directed at students and
rhetorical. These questions, along
with her statements, modeled strategies that students could use to help develop
their proofwriting skills. While
students were not observed to have adopted any of the modeled behaviors, I
believe that these teaching techniques hold promise for changing instruction
and improving student learning.
Classroom Interactions and Proof in an
Exploratory College Geometry Class
Susan Generazzo University of New Hampshire 
Current research indicates that classroom
discourse can have a significant impact on the ways students make sense of mathematical
proof. Students are known to struggle with proof construction and proof understanding, skills that are particularly
important at the college level. This study looks at classroom dynamics in a
college level geometry class of preservice teachers. Data was obtained by
videotaping and audiotaping classroom observations, and by interviewing groups of students. This presentation focuses on interactions between the instructor and the class as they
construct proofs. Excerpts of classroom
dialogue and preliminary coding and analysis of data will be
shared. A framework for data analysis will be based in part on instructional scaffolding and
characterization of utterances described by Blanton et al. (2009). Data analysis
will also build on the ERE (elicitation, response, elaboration) pattern
observed by Bowers & Nickerson (2001), which stems from the IRE (initiation,
response, evaluation) pattern defined by Mehan (1979). Participants are invited to critique the analythe instructor
during the selected episodes.
Jim Gleason The University of Alabama 
The implementation of online texts, videos,
homework, and tests has changed the process of instruction in introductory
college mathematics courses. With this change, more student
learning is taking place outside of the traditional college classroom and in
places such as tutoring centers and dorm rooms. This study explores how these
changes change the impact of the size of the classroom portion of the learning
experience on student involvement in the learning process, instructor
interaction with and feedback to students, and studentsŐ academic performance.
A mixed ANOVA design is used to analyze data generated from College Algebra and
Applied Calculus courses with class sizes ranging from 37 to 129 with common
syllabi, homework, quizzes, and tests.
Student Misconceptions of the Language of
Calculus: Definite and Indefinite Integrals
William L. Hall, Jr. University of Maine 
Many
mathematical terms are also used in everyday English. We say things like origin, derivative, sum, tangent and we mean very specific
things when we are inside a mathematics classroom. The problem here is that when
we step outside a mathematics classroom, these words take on a whole new life;
sometimes they mean the very same thing, and sometimes they are entirely
different entities. In this study, twenty five students in an introductory
calculus course were interviewed about their knowledge of integration.
Participants were asked to discuss various integration problems, both definite
and indefinite, as well as defining the terms "definite integral" and
"indefinite integral." Students provided many different kinds of
responses, but most interestingly, a handful of participants brought up the
point that the definite integral is more "precise" than the
indefinite integral and the indefinite integral is "vague."
Additionally, one student when asked what an indefinite integral was, responded
"I don't know, opposite of a definite integral, obviously." These
types of responses are indicative of not only poor understanding of
mathematical concepts, but also conflict between the students' knowledge of
mathematical terms and their everyday English counterparts.
An
ActorOriented Perspective on Teaching and Learning Mathematical Physics
Mark P. Haugan Purdue University 
Traditional physics instruction, like the traditional
transfer paradigm, is based on the metaphor of application. This is
problematic because, as in LobatoŐs transfer studies, students have difficulty
reasoning about the relationships among measurable quantities in physical
situations and, thus, difficulty applying relevant mathematical knowledge to
physics.
Reformed physics instruction
which focuses on building models to predict or explain the behavior of physical
systems in their surroundings is based, like LobatoŐs
reconceptualization of transfer, on the metaphor of construction. The
PirieKieren model of the growth of mathematical understanding, informed by
EllisŐ recent research on the interplay between generalizing and justifying,
allows us to present one account of how students construct formalized physics
knowledge and knowledge of underlying mathematical structures in the course of
such instruction. This account is
offered for criticism to foster discussion of opportunities for productive
collaboration between the physics and mathematics education research
communities.
No
Teacher Left Behind: Assessment of Secondary TeachersŐ Content and Pedagogical
Content Knowledge
Shandy Hauk WestEd 
Kristin Noblet University of Northern Colorado 
Billy Jackson University of
Northern Colorado 
The
article provides results of five iterations over three years in developing a
written assessment of the mathematical content and pedagogical content
knowledge (PCK) of middle and high school mathematics teachers. Of the 100
teachers to complete written items, half were already "Highly
Qualified" according to No Child Left Behind Act of 2001 criteria and half
were not. Content knowledge items addressed essential understandings for number
and operations, algebra and functions, and proof. PCK measures included subscores
on curricular content, syntactic, anticipatory, and classroom action knowledge.
Results indicate that a "Highly Qualified" teacher with robust PCK
may benefit most from professional development that explicitly addresses
building mathematically rich anticipatory and classroom action knowledge in
addition to opportunities to enrich mathematical vocabulary and content
understandings.
Teaching
Assistants and MidTerm Feedback from Students
Shandy Hauk West Ed shauk@wested.org 
Nasir Awill University of Northern Colorado 
Nissa Yestness University of
Northern Colorado 
As
part of a larger study around the development of pedagogical content knowledge
for college mathematics instruction, this preliminary report explores how
graduate student teaching assistants (TAs) anticipate and adjust to midsemester
feedback from their students. We interviewed three TAs just before they
administered a midcourse evaluation in their classes and did followup
interviews as they went through the completed forms and thought aloud about
their immediate and considered responses to student comments. The goal of the
work is to develop a guide, including a selection of midterm evaluation forms,
for novice college mathematics teachers to use to get the most out of
soliciting and reviewing student feedback.
Self Efficacy and
Mathematical Proof: Are Undergraduate Students Good at Assessing Their Own
Proof Production Ability?
Paola Iannone School of Education and Lifelong Learning University of East Anglia 
Matthew Inglis 
The aim of this research in progress is to
investigate how university students assess their own proficiency in producing
mathematical proofs and how this compares to their actual performance in proof
tasks. There is strong evidence in the educational psychology literature that
selfefficacy is an accurate predictor of academic achievement and the aim of
this small study is to investigate selfefficacy in terms of undergraduate
studentsŐ proof production. The particular focus on proof is of relevance for
its implications for our understanding of undergraduate studentsŐ perception of
what constitutes an acceptable proof. If the findings indicate that
selfefficacy is not a good predictor of proof production then we could argue,
in line with the current literature, that this is a consequence of the
studentsŐ misinterpretation of what they are required to do when asked to write
a proof. The study so far consists of a twoparts questionnaire administered to
72 undergraduates in a university in the UK. The first part is a series of
questions (using fivepoint Likert items) from standard selfefficacy
questionnaires, complemented by questions aimed more directly at proof ability.
The second part consists of four proof tasks, which the students are asked to
complete. We report on the findings from this questionnaire and the
implications of the results on studentsŐ understanding of what is a
mathematical proof. Implications for teaching are also discussed.
Language, Semantic Contamination and
Mathematical Proof
Matthew Inglis Mathematics Education Centre Loughborough University 
Juan Pablo MejiaRamos Graduate School of Education Rutgers University 
The way words are used in natural language can
influence how the same words are understood by students in
formal educational contexts. Here we show that this socalled semantic
contamination effect plays a role in determining how students engage with
mathematical proof, a fundamental aspect of learning mathematics. Analyses of
responses to argument evaluation tasks suggest that students may hold two
different and contradictory conceptions of proof: one related to conviction,
and one to validity. We demonstrate that these two conceptions can be
preferentially elicited by making apparently irrelevant linguistic changes to
task instructions. After analyzing the occurrence of ŇproofÓ and ŇproveÓ in
natural language, we report two experiments that suggest that the noun form
privileges evaluations related to validity, and that the verb form privileges
evaluations related to conviction. Implications of this finding for the
linguistic content of universitylevel assessment materials are discussed.
Cognitive and Emotional
Aspects of Mathematics UndergraduatesŐ Experience of
Visualization in Abstract Algebra
Marios Ioannou University of
East Anglia 
Elena Nardi University of East Anglia 
Abstract
Algebra is considered by students as one of the most challenging topics of
their university studies. Our study is an examination of the cognitive, social
and emotional aspects of mathematics undergraduatesŐ learning experience in
Abstract Algebra. Our data consists of: observation notes and audiorecordings
of lectures and group seminars of a Year 2 course in the UK; student and
lecturer interviews; and, coursework and examination papers. Data analysis is
currently in progress. For the purposes of this paper, following some of our
preliminary observations on the studentsŐ apparently diminishing engagement over
the ten weeks of the course—and, particularly, their comments on the
effect that the abstract, not easily visualizable nature of Abstract Algebra
has on their relationship with the topic—we scrutinize the data sources
listed above for evidence of their perceptions about/attitudes
towards/employment of visualization in Abstract Algebra.
MathematiciansŐ
Mathematical Thinking for Teaching: Responding to StudentsŐ Conjectures
Estrella Johnson, Sean Larsen, and Faith Rutherford Portland State University 
As part of a project involving an inquiry
oriented abstract algebra curriculum we have observed differences in how the
mathematicians using the curriculum responded to student conjectures. Our goal is to characterize these
different responses and to explore possible connections between these
pedagogical moves and the teachersŐ mathematical knowledge for teaching. To do so, we are analyzing
videorecordings data taken during the class sessions to identify and
categorize types of responses given by teachers. Our efforts to explore
connections between these responses and the mathematiciansŐ mathematical
knowledge for teaching are supported by coordinated analyses of the teaching
episodes and debriefing sessions conducted with the mathematicians.
Novice
Teacher Reflections: A
Case Study of Novice TeachersŐ SelfAnalyses of Videos of Their Own Teaching
Rachael Kenney Purdue University 
The
purpose of this study is to look at undergraduate mathematics education majors
engaged in their first full teaching experience and to examine what they attend
to most when watching videos of their own teaching. The preservice teachers in
this study taught a College Algebra course at a large university and met in a
seminar after each lesson to discuss pedagogical and mathematical concerns.
Teachers were asked to videotape one lesson and to watch the video with the
researcher, who recorded comments made by the teachers as they analyzed their
own teaching. An examination of the collected data from this activity can identify the issues in
the classroom that are most important to novice teachers and can inform efforts
to teach teachers how to reflect on their practice.
Using
Advising and Enrollment Data to Inform a FirstYear Math Placement Program
Kristin King University of Northern Colorado 
Joe Champion University of Northern Colorado 
We report on a threeyear project to make
datadriven improvements in the mathematics placement process at the University
of Northern Colorado. We began by analyzing Fall 2007 placement recommendations
for a sample of N=1,466 firstyear
students to the university. These recommendations came from brief
facultystudent interviews during summer orientation sessions in which math
instructors suggested one or more courses for students based on their most
recent mathematics course and grade, high school grade point average, ACT math
score, college major, and other information. We compared these recommendations
to advising and enrollment data over the subsequent year, and, using logistic
regression modeling, identified the background variables that best modeled
success in studentsŐ first mathematics courses. This led us to make changes in
the math placement process for Summer 2009. We describe the new placement
guidelines and summarize preliminary findings from a followup study on the
impact of the changes.
Using
Textbook Projects to Encourage InquiryBased and Collaborative Learning in Multivariable Calculus—A
Teaching Experiment
Brynja Kohler 
Robyn Krohn 
April Lockwood 
Many
mathematics courses on university campuses are largely lecture based. However,
the research suggests that students learn better through inquirybased
instruction and through collaboration with peers. The time commitment to
prepare such tasks is often a problem for professors. To encourage collaborative and inquirybased learning in a
multivariable calculus class, this study will document the results of
implementing projects available in Calculus Concepts and Contexts, by James
Stewart chapters 11 Partial Derivatives, 12 Multiple Integrals, and 13 Vector
Calculus. Student groups will work
on the projects centered on applications or mathematical investigations given
at the end of each chapter in the Stewart Calculus textbook. A rubric will be
used to assess the studentsŐ work, and student surveys will be gathered to determine
the amount and type of collaboration among students as well as the overall
student perceptions of the project.
David Kung 

Most teachers agree that if a student understands a
particular mathematical topic well, she will be able to do problems correctly.
The converse, however, frequently fails: students who do problems correctly
sometimes still hold significant misconceptions about the topic in question. In
this paper we explore this phenomenon in the context of power series, one of
the most challenging topics in the Calculus curriculum. We report on clinical
interviews with students, many of whom arrive at
correct answers to questions about series, explaining their answers in
appropriate terms, despite having significantly flawed ideas about those
series. Implications for teaching power series, other Calculus topics, and
undergraduate mathematics in general are discussed.
Between
Construction and Communication: What Happens During Proof Revision?
Yvonne Lai Department of Mathematics 
Keith Weber Rutgers University Graduate School of
Education 
After work on a mathematics problem, one might (be asked to) communicate a solution. Communicating mathematical reasoning is central to mathematical practice. Yet, what does it mean to communicate proof? What processes occur between proof construction and communication? What constitutes clarity? Answers to such questions can guide instructors of proofbased courses. We report analysis based on data from 10 practicing mathematicians. We presented each mathematician with a statement that uniformly took under a minute to validate as true, yet on average more than 10 minutes to finish writing its proof. We then asked each mathematician to revise proofs of a statement that again was validated swiftly yet took care to communicate. We propose an framework for the processes behind and characterization of clear mathematical communication. The results shed insight into the communicative goals of proof presentation and highlight important aspects of proof that can be emphasized to achieve these goals.
Over the last few decades, researchers have
consistently found that (a) enrollments in remedial mathematics courses at
fouryear universities are increasing, (b) African American and Latino students
are disproportionately enrolled, and that (c) remedial mathematics courses
uniquely mediate studentsŐ access to advanced mathematics and to the
postsecondary education structure writ large. This report/presentation is part
of a broader study in which equitable access to mathematics is defined as the
convergence of structural/institutional forces and individual agency. Drawing
on recent theoretical perspectives on mathematics socialization and identity,
the presentation is focused on one of the studyŐs main research questions: How
do studentsŐ mathematical identities relate to their current (universitylevel)
and high school mathematics engagement? Based on analyses of student
questionnaires, semistructured student and instructor interviews, and
classroom observations, preliminary findings from the studyŐs case
participants—students who were enrolled in a remedial math course during
the fall 2009 semester—will be shared.
On the Histories of Linear Algebra: The Case of Linear Systems

There
is a longstanding tradition in mathematics education to look to history to
inform instructional design. An
historical analysis of the genesis of a mathematical idea offers insight into
(1) the contexts that give rise to a need for a mathematical construct, (2) the
ways in which available tools might shape the development of that mathematical
idea, and (3) the types of informal and intuitive ways that students might
conceptualize that idea. In this
talk, I will discuss historic contexts that gave rise to considerations of
linear systems of equations and their solutions. In particular, I apply SfardŐs (1991) processobject framing
to these historic contexts with an eye toward those contextual framings that
fostered significant theoretical progress toward the development of modern
linear algebra. I will then
discuss the potential of this analysis to inform the design of instructional
materials for an inquiryoriented linear algebra class.
Student
Outcomes from InquiryBased Learning in Mathematics: A MixedMethods Study
Sandra Laursen, MarjaLiisa Hassi, Rebecca Crane, and
AnneBarrie Hunter University of Colorado at Boulder 
Our mixedmethods study examines the outcomes of inquirybased learning (IBL) in a variety of undergraduate mathematics courses at four universities. Classroom observations of courses designated ŇIBLÓ reveal both similarities and differences in how instructors interpret and implement inquirybased learning, but are nonetheless readily distinguished from traditional courses. Instructional practices are linked to student outcomes as measured by pre/postsurvey items about selfreported learning gains, attitudes and beliefs, and in interviews. Students in IBL courses report higher cognitive and affective gains than do nonIBL students, and attribute these gains to specific aspects of classroom instruction and atmosphere. The nature and extent of gains varies by gender and by student audience, including math majors and preservice teachers.
Addressing Impulsive Disposition: Using
Nonproportional Problems to Overcome Overgeneralization of Proportionality
Kien H. Lim 
Osvaldo Morera 
Impulsive
disposition is an undesirable way of thinking where one spontaneously applies
the first idea that comes to mind without checking its relevance. In this
research, we explore (a) the possibility of helping preservice teachers
improve their disposition, from being impulsive to being analytic, in one
semester, and (b) the effect of using nonproportional situations. This study
involves two sections of a mathematics course for preservice teachers for
Grades 48. The lessons were designed whenever possible to illicit studentsŐ
impulsive disposition so that they could become cognizant of it and make
conscious attempts to overcome it. Some test items were designed to be
superficially similar but structurally different to those they had experienced
in class or homework. Prepostend test results show that preservice teachersŐ
tendency to overuse ratios and proportions can be reduced in one semester and
that the use of nonproportional problems can minimize impulsive responses.
Exploratory
Case Study on Negotiation of Mathematical, Pedagogical, and Curricular Meaning
Between Mathematics Education Researcher and Teacher for Professional
Development
This report is a contribution to ongoing research
in teacher professional development.
The main objective of this emergent study is to investigate an ongoing
negotiation of mathematical, curricular and pedagogical meanings between a
mathematics education research and a secondary school teacher as they develop
an Algebra II curriculum. This case study can be an example of needed insight
as to why teachers find it difficult to incorporate and implement alternative
images of classroom practices as a result of working with mathematics education
researchers and as to why researchers find it difficult to communicate these
images as well as understand the felt constraints teachers have in their
teaching practices. This case study can lead into the improvement of
teacher profession programs, the improvement of preservice teacher
preparaactive participation of teachers in classroom experiments.
An
Investigation of PostSecondary StudentsŐ Understanding of Two Fundamental Counting Principles

The addition and multiplication principles are
foundational concepts in counting. Indeed, the appropriate use of these
concepts is fundamental to the successful solution of a wide range of counting
problems, from the very basic to the highly sophisticated. While some existing
research indirectly addresses studentsŐ
uses
of these principles, little has been done to explicitly describe studentsŐ knowledge of these important principles. Adopting Hiebert and LefevreŐs (1986) model for
mathematical knowledge, which characterizes conceptual knowledge as being
marked by connections and relationships among concepts, this research seeks to
provide insights into the ways in which students understand and apply the
addition and multiplication principles as they solve a variety of counting
problems.
A Transition Course From Advanced Placement to
College Calculus
Timothy A. Lucas 
Joseph Spivey 
A growing number of students are enrolling at
universities with AP credit for Calculus I. This results in Calculus II classes
with two very different groups of students, i.e., freshmen and upperclassmen.
It is difficult to construct a Calculus II course that caters to the disparate
needs of these two groups of students. Mathematics departments across the
nation are also debating reform Calculus versus traditional Calculus. The
compromise at Duke University is that two Calculus II courses are offered: (1)
a Laboratory Calculus course that contains many elements of reform Calculus and
(2) a more traditional course. This presents a confusing choice to incoming
students. In the Spring of 2007, a group of highly
motivated graduate students conducted a review of DukeŐs Calculus curriculum.
As a result, this committee carefully crafted a Calculus II course that would
address the needs of incoming students with AP credit and bridge the gap between
traditional and reform calculus. We will present these issues, our proposed
solutions, our experience with running experimental sections of this course and
its future in the Duke mathematics curriculum. This talk may be of interest to
faculty or graduate students who want to review calculus courses at their own
institutions.
Strong Metaphors for the
Concept of Convergence of Taylor Series
Jason H Martin
Arizona State
University 
Michael Oehrtman Arizona State University 
We
present results from questionnaires and interviews that were conducted with
university calculus, real analysis, and numerical analysis students in an
effort to characterize their conceptions of the convergence of Taylor series.
During a detailed analysis of the interviews, we discovered that several
students consistently relied on a single metaphor throughout several tasks. We
were surprised by the studentsŐ commitment to these metaphors (emphasis) and
the degree to which they influenced student responses (resonance). In this talk
we will describe some of the metaphors that students used and how they appeared
to both enable and constrain the studentsŐ reasoning.
Blending InquiryBased and
ComputerAssisted Instruction in an Elementary Algebra Course: A QuasiExperimental Study
John C. Mayer 

Joshua H. Argo UAB 

William O.
Bond UAB 


In
an experiment conducted at the University of Alabama at Birmingham in Fall
Semester, 2009, we compare the effect of incorporating inquirybased group work
sessions versus traditional lecture sessions in an elementary algebra course in
which the primary pedagogy is computerassisted instruction. Our research hypothesis
is that inquirybased group work sessions differentially benefit students in
terms of mathematical selfefficacy, content knowledge, problemsolving,
and communications. All students receive the same computerassisted instruction
component. Students are randomly assigned to a treatment (group work or
lecture). Measures, including pre and posttests, are described. Statistically
significant differences have previously been observed in a similar
quasiexperimental study of multiple sections of a finite mathematics course in
Fall Semester, 2008. Undergraduates who do not place into a credit bearing mathematics course take this
developmental elementary algebra course. Many preservice elementary school
teachers place into elementary algebra, thus making this course a significant
component of preparing K6 teachers.
Making
Actions in the Proving Process Explicit, Visible, and ŇReflectableÓ


John Selden New Mexico State University 
Annie Selden New Mexico State University 
This
preliminary report describes the practices of teachers attempting to alleviate
proving difficulties in a voluntary Ňproving skills supplementÓ to an
undergraduate real analysis course. What happened in the supplement and why it
happened are analyzed in terms of the supplement teachersŐ theoretical
perspective concerning actions in the proving process. Their perspective included that the
proving process is a sequence of actions some, of which are not visible or are
difficult to recall, and that understanding the justification for an action
differs from a tendency to execute it autonomously. Also, the real analysis course and that teacherŐs primarily
lecture based teaching
are briefly described, and a comparison is made between studentsŐ work in the
supplement and their work in the real analysis course. Finally, views of the supplement by the
real analysis teacher, as well as those of three students, are briefly
discussed.
Modeling
the Comprehension of Proofs in Undergraduate Mathematics
Juan Pablo
MejiaRamos Rutgers University 
Keith Weber Rutgers University 
Evan Fuller Montclair State University 
Aron Samkoff Rutgers University 
Robert Search Centenary
College 
Kathryn Rhoads Rutgers University 
Although
proof comprehension is fundamental in higherlevel undergraduate mathematical
courses, there has been no research on what exactly it means to understand a
mathematical proof at this level and how such understanding can be assessed. In
this preliminary report we address these issues by presenting a
multidimensional model of proof comprehension and illustrating how each of
these dimensions can be assessed. Building on Yang and LinŐs (2008) model of
reading comprehension of proofs in school geometry, we contend that in
undergraduate mathematics a proof is not only understood in terms of the
meaning, operational status and logical chaining of its statements (as Yang and
Lin delineate), but also in terms of its higherlevel ideas, the methods it
employs, or how it relates to specific examples. We illustrate how each of
these types of understanding can be assessed in the context of a particular
proof.
Novice College Mathematics InstructorsŐ Teaching Preparation and
Teaching Activities
Bernadette
Mendoza Brady University of Northern Colorado 
This report is on a mixedmethods study
investigating college mathematics instructors' (CMIs') perceptions of teaching.
As part of a multiyear research and development project on CMI experiences, we
administered a webbased College Mathematics Instructor Professional
Development Questionnaire (Hauk, Speer, Kung, & Tsay, 2006). Invitations to all instructors and
teaching assistants were disseminated through the 177 PhDgranting mathematics
departments in the United States. The focus of this report is exploring novice
CMIs selfreporting on three of the subconstructs in the survey: teaching experience, teaching
preparation, and teaching activities. In addition to the quantitative analysis,
I will report on interviews with novice CMIs about the three constructs.
A
Confucian Approach to Teach Algorithms in PreService TeacherŐs Program in the
United States

As
the influential international studies (e.g., TIMSS, 1999, 2003, 2007; PISA,
2003, 2006) showed that Asian students outperformed their American
counterparts, Eastern methods of teaching and learning have attracted much
attention from Western researchers (Stigler & Hiebert, 1999; Fan, Wong,
Cai, & Li, 2004). This preliminary report aims to investigate how a
Confucian teaching approach benefits preservice teachersŐ learning of
algorithms in the United States. The advantages and disadvantages of the
Confucian approach will be investigated theoretically and empirically. A
theoretical analysis will focus on the lessondesign comparisons between the
Confucian Approach and the three constructivist approaches (Simonâ€™s HLT,
1995; KirshnerŐs Construction metaphor, 2002; Lesh & YoonŐs HLT, 2004).
Twelve preservice teachers will be selected from two sections of the same
course for focus group interviews to gain the empirical data. A synthesis
analysis will examine the transportability in adopting the Confucian approach
in the U.S. for preservice teachersŐ learning.
Which Path to Take? StudentsŐ Proof Method Preferences
Melissa Mills Oklahoma State
University 
Students in their first proof course
often lack flexibility while writing original proofs. Learning to utilize
different proof schemes is a crucial factor in their development as
mathematicians. In this preliminary
report, we will consider several student interviews in which students both
think through existing proofs and attempt to generate original proofs. These interviews of beginning abstract
algebra students highlight their capacity to explore multiple ways to prove the
same statement.
The Role of Quantitative
and Covariational Reasoning in Developing Precalculus StudentsŐ Images of Angle Measure and Central
Concepts of Trigonometry
The
presentation will report results from an investigation of three precalculus
studentsŐ conceptions of angle measure, radian as a unit of measurement, and
trigonometric functions. The subjects of the study were enrolled in a
precalculus course using researchbased curriculum that focused on developing
quantitative and covariational reasoning abilities, as well as other
understandings deemed foundational to trigonometry. Results from this
investigation revealed that ideas of angle measure and the radian are
foundational for developing coherent understandings of trigonometric functions.
Specifically, these ideas were necessary to develop coherent meanings and
effectively reason about the geometric objects of trigonometry (e.g., right
triangles and the unit circle) in relation to trigonometric functions. It was
also shown to be important that the students conceived of measureable
attributes (e.g., quantities) of situations and the meaning of the units used
to measure these attributes (e.g., radians) before reasoning about and
formalizing relationships between covarying quantities.
From Beans to Polls: Does Understanding
of Statistical Inference Within a Known Population Context Transfer to an Unknown
Population Context?
Jennifer Noll, Sonya Redmond, and Jason Dolor Portland State University 
In
todayŐs society, informed citizenship requires at least an informal
understanding of statistical inference.
One strategy to promote such understanding is to develop studentsŐ
knowledge of sampling distributions through simulation of repeated sampling
from a known population. It is supposed that students will be able to transfer
their knowledge of sampling distributions created from a known population to
ŇrealworldÓ contexts such as public opinion polls. This preliminary report
presents evidence suggesting that such transfer is neither immediate nor
trivial. We will present case studies from a qualitative interview study of
students enrolled in a statistics for teachers course, illustrating some of the
ways in which students who display a robust knowledge of sampling distributions
apply this knowledge to polling scenarios.
PreService Teachers'
Progression Through the Van Hiele Levels of Geometric Understanding

This
study examined the van Hiele level of geometric understanding of elementary and
secondary preservice mathematics teachers, both before and after taking the
geometry course required by their teacher preparation program. Results indicate that prior to the
course, preservice teachers do not possess a level of understanding at or above
that which would be expected of their target students. Upon completion of the course, findings
show statistically significant gains of at least one level in preservice
teachersŐ van Hiele understanding.
Finally, although statistically significant gains were attained, the
magnitude of the gains was not enough to raise the sample populationâ€™s van
Hiele level to that expected of their future K12 students.
The Role of Intuition in
the Development of StudentsŐ Understanding of Span and Linear Independence in
an Elementary Linear Algebra Class
Frieda Parker University of Northern Colorado 
Cathleen
Craviotto University of Northern Colorado 
In
this presentation, I report on preliminary results from my dissertation study
on the role of intuition in studentsŐ learning of span and linear independence
in an elementary linear algebra class. The purpose of my research is to examine
the relationship between the quality of studentsŐ understanding of span and
linear independence with respect to the role of intuition in their learning of
these concepts. This qualitative study is based on the multiple case study
tradition and employs the theoretical perspective of social constructivism.
Methodological issues of importance in this study are how to assess the quality
of studentsŐ understanding and how to evaluate the role of intuition in that
understanding. Findings from this study might inform the development of more
effective teaching practices for span and linear independence.
What
Makes Rico an Effective Mathematics Teacher?
Ana Lage Ramirez Arizona State University 
This
study contributes to the ongoing discussion of what constitutes mathematical
knowledge for teaching. This presentation will report the case study of Rico, a
high school mathematics teacher, who not only developed profound personal
understandings about the mathematics that he teaches, but profound pedagogical
understandings that can support powerful ways of thinking in his students. The
data for this study is being generated in two phases. Data for Phase 1 included
videos of RicoŐs Algebra II course, postlesson reflections, and
selfconstructed instructional materials. Analysis of Phase 1 data corpus will
lead to Phase 2, which will consist of extensive stimulatedreflection
interviews with Rico. Preliminary results reveal that Rico relies on his images
of his studentsŐ ways of thinking to design instruction that is intended to develop
coherent mathematical ideas that carry through the entire Algebra II course.
RicoŐs curriculum is not about topics or sections from the book. Rather, it is
about helping students to develop ways of thinking that allow them to
internalize the curriculum as a coherent body of ideas.
Katherine S. Remillard Saint Francis University 
Through the lens of smallgroup discourse, this
study explored the learning of mathematical proof by freshman and sophomore
mathematics majors. SfardŐs (2008) conception of Ňthinking as communicating,Ó or commognition,
provided the theoretical framework for the study. The presentation will
highlight the potentiality of the commognitive framework for research in the
area of learning mathematical proof. Specifically, I will give examples of how
Sfard and KieranŐs (2001) complementary discourse analysis tools of focal
(objectlevel) analysis and preoccupational (metalevel) analysis revealed
intricacies of studentsŐ thinking and learning, inaccessible from their work
alone. Moreover, I will discuss how
the analyses of the smallgroup discourse exposed the existence of naturally
occurring points in studentsŐ discussions on proof, called discursive entry points, in which expert intervention might
especially profit student learning.
Mathematics
GTAsŐ Question Strategies
Kitty Roach University of Northern Colorado 
Kristin Noblet University of Northern Colorado 
Lee Roberson University of
Northern Colorado 
JengJong Tsay University of
TexasPan American 
Shandy Hauk WestEd 

As
part of an earlier pilot study, we developed a framework for structuring,
identification, and discussion of questions mathematics graduate teaching
assistants (GTAs) ask while teaching. We continued this research through an
exploratory study examining existing video of five calculus instructors who had
differing levels of teaching experience. Using an analytic inductive approach,
we reviewed existing literature for connections to our question categories,
techniques, and strategies. This qualitative study focused on refining our
framework to create a tool for identifying and discussing the questions GTAs
ask while teaching an undergraduate calculus course.
Supporting Proof Comprehension: A Comparative Study
of Three Forms of Presentation
Somali Roy Loughborough
University 
Lara Alcock Loughborough University 
Matthew Inglis Loughborough
University 
In this session we will report on a project
designed to investigate the effectiveness of three different ways of presenting
mathematical proofs to undergraduates. We first discuss the researchbased
design principles which underlie the creation of a
novel method of proofpresentation: computerbased eProofs. We then report on
an ongoing research project investigating the level of comprehension engendered
by different methods of presenting a proof: the standard written proof with (i)
no further explanation, (ii) explanation as might be given by a lecturer in a
standard undergraduate lecture course and (iii) a computerbased eProof.
Using Online Homework and Data Mining to Assess
Student Learning in Mathematics Courses
Michael B. Scott California State University, Monterey Bay 
This preliminary research report investigates
the use of a webbased homework system that supplements PreCalculus,
Calculus, and Mathematics for Elementary School Teachers courses. The
infrastructure of the homework system has expanded into a course management
system used to coordinate every aspect of the course. Expansion of the system
makes possible the collection of rich data streams about individual students.
Mining this data enables longitudinal studies of student learning. The presenter
will discuss preliminary work exploring mathematical performance patterns as students progress from lowerdivision courses into
upperdivision courses.
Student
Justifications for Relationships between Concepts in Linear Algebra
Natalie E.
Selinski San Diego
State University 
Chris
Rasmussen San Diego
State University 
Michelle
Zandieh Arizona State
University 
A
central goal of most any Linear Algebra course is to ensure that students
develop relational understanding of concepts (Skemp, 1987), become proficient
at various techniques, and develop personal and/or formal justifications for
relationships between concepts.
This preliminary presentation addresses linear algebra studentsŐ
personal justifications for relationships through the analysis of
videorecorded, end of the semester problem solving interviews. In particular,
the interview question analyzed in this report prompted students to consider,
given an invertible matrix A, whether 5 different claims are true or false. These claims are formally part of what
many texts refer to as the
Invertible Matrix Theorem (e.g., Lay, 2003). This report will address student responses to two of these claims by highlighting the informal or personal justifications students provided for relating invertibility, linear independence, and span in R^3.
Mary D. Shepherd Northwest Missouri State University 
This is a small scale study that looks at the
effects of getting firstyear college students to apply reading strategies
specifically related to reading mathematical textbooks to their precalculus
textbook. After a pretest
to assess student exposure to the topics, students were asked to read a passage
from their textbook as they normally would and answer some questions, then read
another passage while being encouraged to apply additional reading strategies,
and answer some questions on that section. Students were randomly assigned to
which passage was read first. The results of this study, while preliminary,
will be presented.
The
Impact of Using Multiple Representations on Precalculus StudentsŐ Conceptual
Understandings of Functions
Soo Yeon Shin Purdue University 
The
purpose of this study is to examine individual college studentsŐ uses of
multiple representations of mathematical functions during problem solving and
the resulting impacts on their understanding of function concepts in
precalculus. I will select five students and examine their understanding of
exponential and logarithmic, higher order polynomial, rational, and
trigonometric functions within symbolic, numerical, graphical, and student
invented representational systems. I will observe their work in problem solving
settings to answer the following questions: (a) how do precalculus students
utilize multiple representations of functions during problem solving, and (b)
what types of mathematics problems do encourage students to integrate each
different representation. An analysis of studentŐs work on mathematical tasks
as well as individual interviews will provide insight for attempting to answer
these questions. From the findings, I hope to be able to suggest ways in which
multiple representations can help students better understand functions in
precalculus classes.
Effective
Folding Back via Student Research in the History of Mathematics
Kelli M. Slaten University of North Carolina Wilmington 
This
preliminary study explores the growth of undergraduatesŐ mathematical
understanding resulting from their research of the historical development of
mathematical concepts. Analysis of studentsŐ written reflections relies on a
key feature of the PirieKieren Dynamical Theory for the Growth of Mathematical
Understanding (1994b), the process of folding back.
Conducting research of the historical development of a particular concept
affords students the opportunity to reexamine and deepen their current levels
of understanding about that mathematical concept. StudentsŐ written reflections
provide insight into their selfawareness of what they learned from their
research efforts and whether the process of folding back was effective and
resulted in growth of mathematical understanding.
Does
Mentoring a Graduate Student Effect Student Achievement?
Hortensia
SotoJohnson University of
Northern Colorado 
Kristin King University of
Northern Colorado 
Coralle Haley University of
Northern Colorado 
In
this quantitative study (N = 160), we explored the relationship between
preservice elementary teachers performance on a common geometry final and the
mentoring status of their instructor. A ChiSquared test indicated no
statistically significant differences in the studentsŐ course grades between instructors,
but an ANOVA revealed statistically significant differences on the final exam
scores. Students of instructors involved with mentoring scored higher than
students who completed the course under the direction of instructors not
involved with mentoring. Our findings suggest mentoring may enhance student
achievement and advocate the need to support mentors. These results may be
strengthened via qualitative research techniques.
The
Nature of Engaging Preservice Teachers in Proof Validation: How Do Their Techniques Compare to MathematiciansŐ
Methods?
Hortensia
SotoJohnson University of
Northern Colorado 
Kristin Noblet University of
Northern Colorado 
Sarah Rozner University of
Northern Colorado 
In
an effort to provide multiple opportunities to validate geometry proofs, we
required preservice teachers to determine the validity of alleged proofs
through presentations, on homework exercises, and on exams. Using qualitative
methods, we investigated preservice teachersŐ views of proof validation,
examined their strategies and fidelity to these strategies, and compared their
validation techniques to those of mathematicians. Our findings suggest our
participants found proof validation challenging but welcomed the opportunity to
engage in an activity that was meaningful to their future careers. Our results
also indicate that our participants were faithful to their claimed validation
strategies and shared a number of validation strategies with mathematicians.
Based on our findings we recommend further research via comparative studies in
order to substantiate whether providing students with instances to validate
proofs helps develop validation processes.
Teacher
Change in the Context of a ProofCentered Professional Development
Osvaldo D. Soto UC San Diego 
The
case study reported here examines the development of proof schemes and teaching
practices of one inservice secondary mathematics teacher who participated in an
offsite professional development (pd) for two years. Two sources of data were
examined: video footage of the teacher doing mathematics at an intensive summer
institute and footage of her own classroom teaching. The analysis of proof
schemes (Harel and Sowder, 1998) focuses on proof production during the summer
institute. Development of the teacherŐs teaching practices was also
investigated during the two academic years following each summer institute. The
report includes theoretical connections (using HarelŐs DNR Theoretical
Framework) between developments in the teacherŐs proof schemes and teaching
practices.
Mathematical Discovery: Proofs and Refutations as a
Model for Defining
Craig Swinyard University of Portland 
Sean Larsen Portland State University 
In their 2008 paper, Larsen and Zandieh analyzed
studentsŐ reinvention efforts in an undergraduate abstract algebra course,
recasting LakatosŐ (1976) descriptions of mathematical discovery as a useful
framework for making sense of studentsŐ mathematical activity. Our work adapts
the framework proposed by Larsen and Zandieh to characterize studentsŐ
mathematical activity in the context of reinventing a definition. Additionally,
our data elucidates the studentsŐ use of counterexamples and proof analysis to
reinvent a definition capturing the intended meaning of the conventional ÎµÎ´
definition of limit. The report has the goal of contributing to a local
instructional theory for supporting studentsŐ understanding of the formal
definition of limit.
John Thompson University of Maine 
Warren Christensen North Dakota State University 
As part of work on student understanding of concepts in advanced thermal physics, we explore student understanding of the mathematics required for productive reasoning about the physics. By analysis of student use of mathematics in responses to conceptual physics questions, as well as analogous math questions stripped of physical meaning, we find evidence that students often enter upperlevel physics courses lacking the assumed prerequisite mathematics knowledge and/or the ability to apply it productively in a physics context. These results suggest advanced students are not incorporating calculus and physics into a coherent framework. We have extended our work to include assessment of mathematical concepts at the end of a multivariable calculus course. Results support the findings among physics students that some observed mathematical difficulties are not just with transfer of math knowledge to physics contexts, but seem to have origins in the understanding of the math concepts themselves.
Open,
Online, Homework Help Forums: What are they and how can they be improved?
Carla van de
Sande Arizona State
University 
Eric Weber Arizona State
University 
Frank Marfai Arizona State
University 
Open,
online, help forums with a spontaneous participation structure allow students
to post problemspecific questions that can be viewed and responded to
asynchronously by others who have the experience, time, and willingness to
help.
Investigations
of existing forums for mathematical help show that the nature of these forums
ranges from cheat sites to sites that operate more consistently with
pedagogically sound principles of learning and instruction. We review our
current understanding of online help forums and share analyses of interaction
in a forum intended to nurture a culture in which communication reflects a
conceptual orientation. The etiquette of this forum stems from Rules of
Engagement for Ňspeaking with meaningÓ that were developed in a facetoface
professional learning community of secondary mathematics and science teachers.
Our investigation targets the ways in which forum etiquette is instantiated on
this site and comparisons of interaction on this site and other homework help
sites.
Student ProblemSolving
Behaviors: Traversing the PirieKieren Model
for Growth of Mathematical Understanding
Janet G. Walter Brigham Young University 
Stacie J. Gibbons Brigham Young University 
It is well known that calculus students often
find Taylor Series difficult to understand and use. As part of a larger study
of calculus learning and teaching, students in a university honors calculus
course explored a task designed to facilitate the emergence of Taylor Series.
In this setting, the growth of one studentŐs understanding of Taylor Series is
examined through the lens of the PirieKieren Model for Growth of Mathematical
Understanding. We focus on identified student behaviors to trace how this
particular studentŐs problemsolving behaviors traversed levels of
understanding as described in the PirieKieren model. Certain student behaviors, which
foster understanding of difficult mathematical concepts
are identified and implications for pedagogical decisions in mathematics
classrooms are noted.
Megan Wawro San Diego
State University 
George Sweeney San Diego
State University 
Jeffrey M.
Rabin University of
Califormia San Diego 
The
purpose of this study is to investigate student conceptualizations of subspace
in linear algebra. In interviews
conducted with eight undergraduates, we found that the imagery for subspace
offered by students varied substantially from the conceptŐs formal definition.
This is consistent with literature in other mathematical content domains
indicating that a learnerŐs primary understanding of an idea is not necessarily
informed by that ideaŐs formal definition. We used the notion of concept image
and concept definition (Tall & Vinner, 1981) in order to highlight this
distinction in the student responses. Through grounded analysis, we identified
recurring concept imagery, such as: geometric or algebraic object, containment,
and division. Furthermore, we found that many students interviewed expressed
the common—and
mathematically imprecise—notion that Rk is a subspace of Rn for k
< n. We conclude by reflecting on
how our findings can inform teaching.
The
Role of Precalculus Curriculum that Emphasizes the Development of Quantitative and Covariational Reasoning on StudentŐs
Ability to Model and Solve Novel Problem Tasks
Eric D. Weber Arizona State University 
Marilyn P. Carlson Arizona State University 
Quantitative
and covariational reasoning are critical reasoning abilities for
conceptualizing and representing quantities in an applied problem context
(Carlson, Jacobs, Coe, Larsen, & Hsu, 2002; Carlson, Larsen, & Jacobs,
2001; Saldanha & Thompson, 1998; Thompson, 1994). This investigation into
precalculus studentsŐ quantitative and covariational reasoning ability intended
to reveal new insights into how the development of these ways of thinking affect studentsŐ solving of applied precalculus
problems over the course of one semester.
Six precalculus students were tracked over the course of a semester
interacting in the classroom, completing research based homework tasks in small
groups, and participating in clinical interviews. The goal of this study was to
reveal the role students construction of quantity, and covariation of
quantities impact play in studentsŐ orientation to novel problem solving tasks.
The Pedagogical Practice of Mathematicians: Proof
Presentation in Advanced Mathematics Courses
Keith Weber Rutgers University 
This talk investigates two areas of undergraduate
mathematics that are in need of research: how proofs are presented in
mathematics courses and the teaching practices of mathematics professors.
Interviews with nine mathematicians explored the issues of what they presented
proofs to students and what they expected students to learn, how they presented
proofs, how they expected students to read proofs, the ways in which they
assessed students' understanding of a proof, and whether proof was appropriate
for all students. Although the participants' goals for presenting proofs were
generally consistent with those of the mathematics education community, some
participants reported that they lacked the pedagogical strategies for
presenting proofs effectively or assessing students' understanding of proof.
Undergraduate
StudentsŐ Interpretations of the Equals Sign
Aaron Weinberg Ithaca College 
While
many instructors may assume that their students have a good conception of
equality, recent investigations on studentsŐ algebraic reasoning suggest that
this may not be the case. This report examines the ways undergraduate students
interpret expressions involving the equals sign and use the equals sign to
represent situations involving comparisons. The study uses a semiotic and
situated perspective to discuss the ways students ascribe meaning to the equals
sign and its implications for instruction.
Symbol
Sense in Linear Algebra: Student Interpretations of A[x y]=2[x y]
Michelle
Zandieh Arizona State
University 
Francesca
Henderson San Diego
State University 
Chris
Rasmussen San Diego
State University 
Megan Wawro San Diego
State University 
George Sweeney San Diego
State University 
Christine
Larson Indiana
University 
In
this report we detail linear algebra studentsŐ interpretation of an equation
that is fundamental to eigen theory, prior to any
formal instruction on eigenvalues and eigenvectors. Data for this analysis
comes from semistructured problem solving interviews with 13 undergraduate
students as part of a semesterlong classroom teaching experiment in linear
algebra. We identified three main categories of student reasoning about the two
equations: 1) students who used superficial algebraic cancellation, 2) students
who correctly solved the system but were unable to interpret their result, and
3) students who correctly solved the system and correctly interpreted their
result. More broadly, the research reported here lays the groundwork for
characterizing student symbol sense in linear algebra.