RUME XVII CONFERENCE SCHEDULE
Thursday, February 27, 2014
1:00 – 1:15 pm Grand Mesa DEF 
Opening Session

1:25 – 1:55 pm 
Session 1 – Preliminary Reports 
Chasm Creek A 
Characteristics Of Successful
Programs In College Calculus At Bachelors Granting Universities Kathleen Melhuish,
Sean Larsen, Erin Glover & Estrella Johnson The CSPCC (Characteristics of Successful Programs in College
Calculus) project is a large empirical study investigating mainstream
Calculus 1 to identify the factors that contribute to success, and to
understand how these factors are leveraged within highly successful programs.
Phase 1 of CSPCC entailed largescale surveys of a stratified random sample
of college Calculus 1 classes across the United States. From these surveys,
successful institutions were selected as case studies. At each case study
institution, Calculus I instructors, students and relevant administration
were interviewed. In this report, we will present preliminary analysis on the
five bachelors granting institutions selected. We will discuss common themes
and factors that have emerged from the five institutions. 
Grand Mesa A 
Students Understanding Of
Exponential Functions In The Context Of Financial Mathematics Natalie E. Selinski Exponential functions are one
of the most critical mathematical topics used by students in financial
mathematics. This presentation explores university finance students notion
of exponential function from two sets of data. First, I use data collected
through surveys to examine students understanding of exponential function in
general and, more specifically, to identify the extent to which students
conflate exponential functions with polynomials. I then draw on data
collected in an inquirybased instructional sequence aimed at improving
financial mathematics students understanding of exponential functions.
Results include delineation of what ways of understanding exponential
functions are critical to studying financial mathematics and insights into
how best to guide students in developing these understandings within the
context of their field of study. 
Chasm Creek B 
Diagrams In Advanced Mathematics: Affordances
and Limitations Kristen Lew, Tim
FukawaConnelly, Juan Pablo MejiaRamos & Keith Weber We report a case study aimed at
researching the rationale of a university mathematics professor for using
diagrams in his analysis lectures, what he hoped his students would learn
from these diagrams, the ways students understand these diagrams, and what
they learn from them. Preliminary analysis suggest that by focusing on
specific properties of the diagrams presented in mathematics lectures, or by
attributing little importance to them, students fail to fully understand what
professors hoped they would learn from these diagrams. 
Grand Mesa B 
Exploring Students Questions From
Online Video Lectures Fabiana Cardetti,
Konstantina Christodoulopoulou & Steven Pon This study was designed to
investigate the types of questions college students generate as they watch
video lectures in a business calculus class. Thirtysix students taking an
undergraduate calculus course participated in the study. In this paper we
share the preliminary results of our qualitative analysis. We have found nine
mutually exclusive categories that uncover the thoughts, struggles, and successes
our students go through as they experience this new teaching modality of
videoviewing. We also include three questions for the audience to help
further our analysis and open up new research opportunities for the
improvement of collegiate teaching through the study of students questions. 
Grand Mesa C 
Transfer Of Learning: Examining
Individuals In Social Settings Jeffrey King,
Stephenie Anderson & Gulden Karakok In this preliminary report, we
share the design and results of the first phase of our ongoing research
study. Our threephase study is designed to investigate individual students
transfer of learning of linear algebra concepts along with social
mathematical interactions in which such concepts developed in groupbased
courses. We first frame our study in relation to current literature, then
discuss our initial analysis from the first phase. Finally, we give a
description of upcoming phases along with questions we wish to discuss with
the audience. 
2:05 – 2:35 pm 
Session 2  Contributed Reports

Chasm Creek B 
Geometric Reasoning On The Complex
Plane Hortensia
SotoJohnson & Jonathan Troup Using Bakker and Hoffmans
(2005) framework on diagrammatic reasoning, we analyzed a videotaped
interview to explore two undergraduates ability to reason geometrically
about tasks related to complex variables. Our findings indicate that in order
to provide a geometric interpretation, our participants needed to first
perform algebraic computations. These computations appeared to provide them
with the pieces required to construct a diagram. Once these pieces were in
place the participants used dynamic gesture to enact their geometric
interpretations with the aid of their diagram. It appeared that their dynamic
gestures assisted with embodying geometric interpretations and as such one
particular task was influential throughout the interview. Furthermore, the
participants integrated less dynamic gesture as they progressed with similar
tasks. 
Grand Mesa A 
Preservice Elementary Teachers'
Understanding Of Number Theory: Connecting Content Knowledge To Pck Kristin
Noblet Little is known about the
relationship between preservice elementary teachers content knowledge and
PCK (Shulman, 1986), especially in number theory. This was investigated as
part of a larger case study of preservice elementary teachers understanding
of topics in number theory. An emergent perspective (Cobb & Yackel, 1996)
as well as Ball and colleagues' conceptualization of Mathematical Knowledge
for Teaching (e.g, Hill, Ball, & Schilling, 2008) were used to collect
and analyze data in the form of field notes, student coursework, and
responses to taskbased oneonone interviews. The study suggests that
preservice elementary teachers' PCK can be strengthened and influenced by
their specialized content knowledge (Ball, Thames, & Phelps, 2008) as
well as their perspectives on how students learn. 
Grand Mesa B 
Students Examples Usage In The
Domain Of Functions Muhammed Fatih
Dogan Mathematicians use examples strategically while working on
mathematical conjectures, and this strategic usage helps them gain a lot of
insight about mathematical phenomenon. However, students do not always have
the same strategic example usage; instead, they tend to over rely on examples
without understanding of example based reasoning. This study examines college
algebra students responses on a written assessment in the function domain
and discusses students example spaces. The results reveal that students have
very limited example space in the function domain that affects their
strategic example usage. Student example usage was very limited to
conventional example spaces that they learned during instruction or from
their textbook. This study suggests that having conventional example spaces
does not guarantee that students can use examples strategically which can
help them better understand the mathematical conjectures. 
Chasm Creek A 
Perceptions In Abstract Algebra:
Identifying Major Concepts and Concept Connections Within Abstract Algebra Ashley
Suominen Abstract algebra is recognized as a highly problematic course for
most undergraduate students. Despite these difficulties, most mathematicians
and mathematics educators affirm its importance to undergraduate mathematical
learning. Both the process of abstraction and constructing on past knowledge
are essential to comprehending course material. The goal of this research was
to establish a list of the important concepts in abstract algebra as
perceived by graduate students in mathematics and understand how they believe
these concepts are related. Through an interview study, the students
perceptions of Abstract Algebra were analyzed through the development of
concept maps. The results revealed graduate students had great difficulty
articulating what they learned and had differing views of major concepts and
relationships within the course. Furthermore, their perceptions of concept
importance equated to the amount of time their class spent discussing that
concept. 
2:35 – 3:05 pm Atrium 
Coffee Break

3:05 – 3:35 pm 
Session 3 – Preliminary Reports

Grand Mesa A 
Calculus Instructors' Resources,
Orientations and Goals In Teaching Low Achieving Students Misun Lee &
Sepideh Stewart Teaching and learning calculus
has been the subject of mathematics education research for many years.
Although the body of research is mainly concerned with students difficulties
with calculus, in this study we will be focusing our attention on the professors
and instructors of calculus. In this research we used Schoenfelds framework
to examine four instructors resources, orientations and goals in teaching
calculus to low achieving students. So far, the preliminary results of the
interviews show that although the professors thought differently about many
aspects regarding teaching calculus, they all claimed that the first step to
succeed in calculus courses is being prepared and having the right
background. 
Chasm Creek A 
Conceptions Of Inverse
Trigonometric Functions In Community College Lectures, Textbooks, and Student
Interviews Vilma Mesa &
Bradley Goldstein We present a textbook analysis
of conceptions of key ideas associated with inverse trigonometric functions
using Balacheffs model of conceptions (Balacheff & Gaudin, 2010). We
found conflicting conceptions of angles, trigonometric functions, and inverse
trigonometric functions that may help explain difficulties that community
college trigonometry instructors and their students face when explaining
tasks associated with this topic. We make suggestions for further research. 
Chasm Creek B 
An Analysis Of TransitionToProof
Course Students Proof Constructions With A View Towards Course Redesign John Selden, Ahmed
Benkhalti & Annie Selden The purpose of the reported
study was to gain knowledge about undergraduate transitiontoproof course
students proving difficulties. We analyzed the final examination papers of
students in one such course. We have tentatively identified categories of
difficulties such as nonstandard language/notation, insufficient warrants,
and extraneous statements. The ultimate goal is to use these categories as
pedagogical content knowledge with which to redesign an existing
transitiontoproof course to alleviate the difficulties for future students. 
Grand Mesa B 
Graduate Students Integrated
Mathematics and Science Knowledge For Teaching Shahram Shawn
Firouzian Previous studies have indicated
that effective mathematics teaching relies on teachers knowledge of both
student thinking and mathematical content, however very little is known about
the integration (combination) of teachers mathematical knowledge and science
knowledge for teaching important topics like derivative and applied
derivative problems. The goal of this study was to examine the knowledge of
mathematics and science that teachers draw on when teaching the concept of
derivative and applied derivative problems. We conducted taskbased
interviews with nine graduate assistants (GTAs). Findings revealed that GTAs
made use of their knowledge of science as well as of mathematics when
discussing how to teach applied derivative problem. In this proposal, we only
look at the results of two interviews and try to shed light into the nature
of science and mathematics knowledge the teachers use for Teaching and how
that can lead into opportunities in professional development for the novice
teachers. 
Wind Star 
Implementing InquiryOriented
Instructional Materials In Undergraduate Mathematics Christine Larson,
Megan Wawro, Michelle Zandieh, Chris Rasmussen, David Plaxco & Katherine
Czeranko Over the past years, research
in the RUME community has driven the development of inquiryoriented
instructional materials in a number of undergraduate mathematics content
areas including abstract algebra, differential equations, and linear algebra.
Literature at the K12 level has documented challenges inherent to scaling up
the implementation of this kind of instruction. In this study, we explore how
instructors make sense of and implement inquiryoriented instructional
materials in undergraduate mathematics, and the nature of supports these
instructors report using and wanting when planning for instruction. We
consider instructors interpretations and desired supports as they relate to
prior pedagogical experience and institutional setting. Data is taken from
surveys, interviews, and videotaped instruction of three participating
instructors at three different institutions as they work to implement two
inquiryoriented instructional units in undergraduate linear algebra. 
Grand Mesa C 
An Observation Instrument For
Assessing The K16 Mathematics Classroom Jim Gleason We describe the development of
a new observation protocol instrument for classroom instruction that is
mathematicsspecific, spans K16 mathematics, improves validity and
reliability compared to existing instruments, and encompasses the Standards
for Mathematical Practice. The instrument may be helpful for educators/researchers
engaged in classroom evaluations of K16 mathematics teaching. 
3:45 – 4:15 pm 
Session 4 – Contributed Reports

Grand Mesa A 
Generalizing Calculus Ideas from
Two Dimensions to Three: How Multivariable Calculus Students Think About
Domain and Range Allison Dorko &
Eric Weber We analyzed multivariable
calculus students meanings for domain and range and their generalization of
that meaning as they reasoned about domain and range of multivariable
functions. We found that students thinking about domain and range fell into
three broad categories: input/output, independent/dependent variables, and/or
as attached to specific variables. We used Ellis (2007) actororiented
generalizations framework to characterize how students generalized their
meanings for domain and range from singlevariable to multivariable
functions. This framework focuses on the process of generalization –
what students see as similar between ideas in multiple contexts. We found
that students generalized their meanings for domain and range by relating
objects, extending their meanings, using general principles and rules, and
using/modifying previous ideas. Our results about how students understand and
generalize the concepts of domain and range imply that the domain and range
of multivariable functions is a topic instructors should explicitly address. 
Chasm Creek B 
How To Make Time: The Relationships
Between Concerns About Coverage, Material Covered, Instructional Practices,
and Student Success In College Calculus Estrella Johnson,
Jessica Ellis & Chris Rasmussen This report draws on data
collected by the Characteristics of Successful Programs in College Calculus
project in order to investigate issues around coverage and pacing. This
includes identifying what topics are being taught in Calculus I, determining
the extent to which instructors and departments feel pressure to cover a set
amount of material, and investigating possible relationships between concerns
over coverage, instructional practices, and the nature of the material
covered at five institutions selected for having successful calculus
programs. 
Grand Mesa B 
Do Experts and Novices Gesture
Differently? Brent Hancock,
Marki Dittman & Hortensia SotoJohnson Previous gesture studies
conjecture that as individuals develop expertise in a field of mathematics
their gestures tend to become more metaphoric, iconic, and dynamic. In this
mixedmethods study, we compared the gestures of six experts and four pairs
of novices as they geometrically described the complex number arithmetic
operations: z+w, zw, and 1/z. An ANOVA revealed that the factors Task and
Gesture were statistically significant, but there was no statistically
significant difference between the two groups gesture use. A Hierarchical
Cluster Analysis directed the qualitative analysis where we found that
novices exposed to technology appeared to produce gestures that were
innovative or similar to the experts gestures. These findings suggest that
bolstering students awareness of their own and the instructors gestures as
well as exposing students to technology may help them develop more dynamic
gestures and in turn possibly facilitate a more geometric perspective of the
arithmetic of complex numbers. 
Chasm Creek A 
Why Lectures In Advanced
Mathematics Often Fail Kristen Lew, Tim
FukawaConnelly, Juan Pablo MejiaRamos & Keith Weber Research on mathematicians
pedagogical practice in advanced mathematics is sparse. The current paper
contributes to this literature by reporting a case study on a proof that a
professor presented in a real analysis course. By interviewing the professor,
we focus on his learning goals in this proof and the actions that he took to
achieve these goals. By interviewing six students, we investigate how they
perceived the proof and what they learned from it. Our analysis provides
insight into why students did not learn what the professor desired from this
lecture. 
4:25 – 4:55 pm 
Session 5 – Contributed Reports

Grand Mesa B 
Working Together On Mathematics
Homework: A Look At How University Students Spend Their Time Outside The
Classroom Gillian Galle Despite the large amount of time university students are expected to
spend studying material and learning on their own outside of the classroom,
little is known about what specific student study habits look like. This
study sought to start developing a description of what activities students
engage in when studying together in selfformed groups outside of the
classroom. By identifying a set of macrotasks, verballycued transactions
that identify what activity the group is currently engaged in doing, this
study provides a way to compare how different study groups allocate their
time and distinguish between the enactment of social and sociomathematical
norms outside of the classroom. 
Chasm Creek B 
A Comparison Of Four Pedagogical
Strategies In Calculus Spencer Bagley The quality of education in
introductory calculus classes is an issue of particular educational and
economic importance. In work related to a national study of college calculus
programs conducted by the MAA, I report on a study of four different
pedagogical approaches to Calculus I at a single institution in the Fall 2012
semester. Using statistical methods, I analyze the effects of these four
approaches on students persistence in STEM major tracks, attitudes and
beliefs about mathematics, and procedural and conceptual achievement in
calculus. Using qualitative methods, I draw links from the statistical
results to differences and commonalities in the four classroom strategies. 
Chasm Creek A 
Implied and Empirical Readers Of
Newtons Method Kristen Murphy,
Celeste Glenn & Nicole Engelke Infante The ability to translate a text
into a mathematical process is a key goal of mathematics education. Knowing
when students have the prerequisite knowledge to understand such a process is
a perennial concern for instructors. Here we use Newtons method to evaluate
reader oriented theory as a means to illuminate these issues. Through
clinical interviews with twelve first semester calculus students, we
determined that knowledge of both tangent lines and roots is required for
students to understand and apply Newtons method. Analysis was done from the
perspective of the empirical, implied, and intended readers and was examined
for the extent to which the empirical and implied readers aligned. It was
found that although the alignment of the empirical and implied readers was
helpful in determining the success of the students, it was not in itself a
deciding factor. 
Grand Mesa A 
Differences In Expectations Between
Explicit Statements and Actual Practices Using Vectors In A Trigonometry and
Physics Course Wendy James Science and engineering
instructors often observe that students have difficulty using or applying
prerequisite mathematics knowledge in their courses. Historically, transfer
theory is used to investigate students issue applying their vector knowledge
from a trigonometry course to a physics course, but this qualitative
casestudy is positioned differently epistemologically and theoretically from
transfer theory to understand and describe the mathematical vector practices
in the two courses. Saussures (1959) concept of signifier and signified
provided a lens for examining the data during analysis. Multiple recursions
of withincase comparisons and acrosscase comparison were analyzed for
differences in what the instructors and textbooks explicitly stated and later
performed as their practices. While the trigonometry and physics instruction
differed slightly, the two main differences occurred in the nature and use of
vectors in the physics course. 
5:00 – 6:10 pm Centennial Room (12^{th} Floor) 
Poster Session


Beyond Plug and Chug: The Nature Of
Calculus Homework At Doctoral Institutions Gina Nunez, Kady
Hanson & Jessica Ellis Prior research reflects a positive relationship between homework and
student academic achievement in undergraduate mathematics courses.
Additionally, recent research has indicated no significant difference in
student learning based upon the medium of the assignment (online based
versus paperbased). These findings led us to ask the question: How does the
nature of Calculus I homework assignments at doctoral institutions with successful
calculus programs compare to assignments at institutions with less successful
calculus programs? Descriptive analyses of student and instructor responses
from a large national survey given to mainstream Calculus I programs were
conducted. Analysis revealed significant differences in the nature of
homework between successful and less successful institutions, including
differences in the content and frequency of assignments. The holistic
approach to homework taken by successful institutions adds to the existing
literature on homework at the undergraduate level and indicates an
interesting relationship between homework and student success in Calculus I
courses. 

Students' and Experts' Ways Of
Reasoning About Partial Derivatives Across Stem Contexts Eric Weber, Tevian
Dray, Corinne Manogue, Mary Bridget Kustusch & David Roundy A common feature across STEM disciplines is the study of change,
whether studying how changing a design parameter affects the operation of a
prototype, or how pressure changes when we adiabatically compress a gas.
Indeed, the nature of scientific measurement is to control some physical
quantities while measuring others. Mathematically, we express the concept of
changing one parameter while fixing others by using partial derivatives.
However, how we use partial derivatives and how we talk about partial
derivatives vary dramatically across STEM disciplines. The purpose of this
poster is to share our preliminary results from student and expert
problemsolving interviews about partial derivatives. 

Formal Logic In Early Undergraduate
Mathematics: A Cycle Morgan Dominy There are certain concepts in early undergraduate mathematics such as
limits and linear independence which heavily rely on student understanding of
formal logic. Since some undergraduates will eventually become preservice
teachers, any changes to early undergraduate courses will have a ripple
effect throughout all levels of education. The purpose of this poster is to
start a dialogue among math educators and gain insight on how to proceed in
conducting research to measure the effect. 

Raising Calculus To The Surface:
Discovering Multivariable Calculus Concepts Using Physical Manipulatives Aaron Wangberg,
Brian Fisher, Jason Samuels & Eric Weber Current research on algebraic and quantitative reasoning shows that
many students experience mathematics as the manipulation of meaningless
symbols (Smith & Thompson, 2007). In order to develop meaning in symbolic
contexts, students must first conceive of relationships between the
underlying quantities present in a particular context. Our project focuses on
a quantitative reasoning approach to multivariable calculus, in particular
the concepts of function, rate, area and volume by using physical surfaces.
In this poster, we provide examples of identifying, measuring, and recording
of essential quantities on physical surfaces. 

Visualizing Mathematical
Connections In Student Teaching Episodes Danielle Champney This poster aims to
present a modified version of SPOT diagrams (Structure Perceived Over Time)
(Yoon, 2012) – an aspect of analysis and data presentation used to
present interactive student video data during which perceptual shifts may
occur. Versions of this analytic tool are being explored with data from a
study of students' teaching episodes during which they explained their
understanding of infinite series convergence. Examining the teaching episode
of one student (Molly) who had a literal aha! moment during her
explanation, the aim of this poster will be to share the affordances of these
modified diagrams, and discuss their benefits for exposing some of the
factors that contributed to Molly's developing understanding of infinite
series convergence, as her teaching episode unfolded. 

Developing Inquiry Oriented
Instructional Materials For Linear Algebra (Dioimla): Overview Of The
Research Project Megan Wawro,
Michelle Zandieh, Chris Rasmussen, Christine Larson, David Plaxco &
Katherine Czeranko The goals of the recently funded DIOIMLA research project are to
produce: (a) student materials composed of challenging and coherent task
sequences that facilitate an inquiryoriented approach to the teaching and
learning of linear algebra; (b) instructional support materials for
implementing the student materials; and (c) a prototype assessment instrument
to measure student understanding of key linear algebra concepts. Our poster
will provide more detailed information about the DIOIMLA research project.
Each of the three aspects of the project will be described in more detail and
examples of each will be shared. The poster will also include an overview of the
current status of the research project and a summary of the timeline for
planned future work. 

Measurement Definitions For
Elementary School Teachers: Links To Graduate Level Mathematics Visala Rani Satyam Undergraduates planning to be teachers often encounter mathematics
content textbooks written specifically for their population (preservice
teachers). Elementary mathematics textbooks of this kind provide indepth
definitions of elementary school mathematics to foster deeper understanding
of these basic concepts. I looked at measurement definitions (length, area,
and volume) across 6 preservice textbooks and identified overarching themes,
using an open coding method. These definitions tend to the precise and
rigorous. For example, Parker & Baldridge (2008) define area as a way of
associating to each region R a quantity Area(R) (p. 107). The following
themes emerged across the set of definitions: discrete/continuous, unit, no
overlaps/full cover, interior/exterior, function, measurement as an
attribute, and space filling. I end with a discussion of the links to
graduate level mathematics and what this means for preservice teachers and
their future elementary students. 

Using a Framing and Resources
Framework For Analyzing Student Thinking About Matrix Multiplication Warren
Christensen A student who has completed both Linear Algebra and Quantum Mechanics
should have a wealth of conceptual and procedural knowledge that has been
obtained from mathematics and physics classes. However in practice, students
seem to struggle with this task. This investigation casts light on students
thinking about matrix multiplication and how their thinking appears to be
influenced by their framing of the problem as either a mathematics or physics
question. Using Framing and Resources as a theoretical lens can provide
insight into the ideas and concepts that a student accesses from domains of
mathematics and physics. Using lexicon analysis, it appears the student
shifts from a mathematical frame to a physics frame and back again, but
struggles to successfully transfer concepts between these two frames. I will
highlight the markers for these frame shifts and demonstrate why framing and
resources is the appropriate lens for this investigation. 

Mathematical Thinking In
Engineering and Mathematics Students Jenna Tague The past decades have brought a multitude of calls for
improving the mathematical education of Science, Technology, Engineering, and
Mathematics (STEM) students as well as increasing the number of STEM
graduates (FerriniMundy & Gler, 2009). However, there is a need
to examine what mathematics and mathematical thinking is needed for these
STEM disciplines. This study examined the mathematical thinking of two
purposefully selected students (one from mathematics and one from
engineering) enrolled at a large Midwestern university as a starting place in
addressing this gap. Interviews were analyzed through the socio cultural lens
of zone theory (Valsiner, 1997) in order to investigate the resources the
students drew upon while thinking mathematically. Additionally, a
mathematical modeling cycle (Blum & Leiβ, 2007) allowed for
cataloguing the particular phases involved in the participants mathematization
processes. 

Pencasts As Exemplars In
Differential Equations Jennifer Czocher, Jenna Tague,
Amanda Roble & Gregory Baker A substantial amount of students time in mathematics
courses at the undergraduate level is spent working homework problems, but it
is difficult to help students make sense of procedures or go beyond acquiring
fluency. We introduced pencasts as an instructional medium to a differential
equations course to create exemplars of challenging homework problems which
cognitively modeled the professors mathematical thinking on each task. Using
mixedmethods survey design, we assessed students use of and response to the
pencasts. Students reported finding the pencasts helpful, that they enabled
independent work on the homework, and they appreciated a detailed, verbal
explanation of why particular steps were taken in solving each exemplar. 

Mathematics Beyond Classroom:
Students' Value Creation Through Mathematical Modeling Within a Learning
Community Joo Young Park This
study examined how mathematical modeling activities within a collaborative
group impact on students perceived value of mathematics. With a unified
framework of Makiguchis theory of value, mathematical disposition, and
identity, the study identified the elements of the valuebeauty, gains, and
social goodwith the observable evidences of mathematical disposition and
identity. A total of 60 college students participated in Lifestyle
mathematical modeling project. Both qualitative and quantitative methods were
used for data collection and analysis. The result from a pairedsamples
ttest showed the significant changes in students mathematical disposition.
The results from analysis of students written responses and interview data
described how the context of the modeling tasks and the collaborative group
interplay with students' perceived value. The poster will present the main
findings and the examples of students written tasks and responses. 

A Proposal
For a Theoretical Framework On Specialized Knowledge For Teaching Mathematics Thorsten Scheiner Building upon past and recent theoretical approaches and models in
research on mathematics teacher knowledge, the presented work provides a
theorydriven and researchbased approach conceptualizing the construct of
specialized knowledge for teaching mathematics. The crucial aspect of this
approach is the underlying assumption that the transformation of knowledge
from specific knowledge bases creates a new form of knowledge that possesses
distinct characteristics that were not present in their original form. This
new kind of knowledge is considered as being crucial for teaching
mathematics, in particular, at an uppersecondary level. 
6:15 – 9:00 pm Grand Mesa DEF 
Dinner & Plenary SessionPlenary
Speaker: Andrea Disessa Knowledge In
Pieces: How To Analyze The Process Of Learning At High Resolution Abstract: I aim to give an overview of how the epistemological
perspective of Knowledge in Pieces (KiP) has allowed the creation of
highresolution analyses of learning in process. High resolution entails
analysis of realtime data, so that one can actually see learning steps at
the grainsize at which learners experience it, and at which teachers and
curriculum developers try to manage it. As such, this very rare kind of
analysis might be extraordinarily helpful in designing instruction and
learning materials. I will first try to characterize the overall KiP program
of studies and contrast it with other programs of studying learning. Then, I
will use data from two recent studies to illustrate the principles in action.
(1) I will show a case of a small class of students developing, on their own,
some normative physics (Newtons law of thermal equilibration: Temperature
difference drives rate of change of temperature). Here, we can see, element
by element, what incoming knowledge was invoked, and how it changed and
combined to result in the normative idea. (2) The other study involves
microanalysis of student learning from a wellstudied instructional sequence
(Brown and Clements bridging analogies). In this, we track differences in
incoming student knowledge well enough to see why some students succeeded and
others failed to achieve the instructional goal.Plenary 


Friday February 28, 2014
8:45 –
9:15 am 
Session 6 – Contributed Reports

Grand Mesa A 
Calculus Students' Early Concept
Images Of Tangent Lines Renee Larue,
Brittany Vincent, Vicki Sealey & Nicole Engelke This study began as an attempt
to explore firstsemester calculus students understanding of Newtons
method. Within that context, it was found that many students had difficulty
sketching tangent lines. The research presented in this paper examined the
language students use to describe tangent lines as well as their graphical
illustrations of tangent lines. Taskbased interviews were conducted with
twelve firstsemester calculus students who were asked to verbally describe a
tangent line, sketch tangent lines for multiple curves, and use tangent lines
within the context of Newtons method. We identified six prominent categories
that described students concept images of tangent lines and found that
individual students often possessed multiple concept images. Furthermore,
data shows that these concept images were often conflicting, and students
were usually willing to modify their concept images in different contexts. 
Chasm Creek A 
Lessons Learned From Case Studies
Of Successful Calculus Programs At Five Doctoral Degree Granting Institutions Chris Rasmussen,
Jessica Ellis & Dov Zazkis In this report, we present
initial findings from our case study analyses at five exemplary calculus
programs at institutions that offer a doctoral degree in mathematics.
Understanding the features that characterize exemplary calculus programs at
doctoral degree granting institutions is particularly important because the
vast majority of STEM graduates come from such institutions. Analysis of over
95 hours of interviews with faculty, administrators and students reveals
seven different programmatic and structural features that are common across
the five institutions. A community of practice and a socialacademic integrations
perspective are used to illuminate why and how these seven features
contribute to successful calculus programs. 
Grand Mesa B 
Mathematics Teacher Models Of
Quantitative Reasoning David Glassmeyer,
Michael Oehrtman & Jodie Novak The purpose of this study was
to document mathematics teachers' models of quantitative reasoning as they
participated in a Model Eliciting Activity (MEA) grounded in their classroom
practice. This MEA was designed and implemented in a master's course of 21
inservice mathematics teachers. The documents produced by the teachers were
analyzed using a models and modeling perspective to determine how teacher
models of quantitative reasoning developed through the MEA. Findings from
this study detail how teachers models of quantitative reasoning were not
fully communicated in terms of defining quantitative reasoning in settings
not connected to their classroom. As teachers went through the course and the
MEA iterations, they began grappling with quantities and quantitative
relationships as aspects of quantitative reasoning. Teachers attention to
these aspects better positioned these teachers to reason covariationally
about the mathematical content in their documents, thus promoting deep
conceptual understanding of functions and more advanced mathematical topics. 
Chasm Creek B 
The Selection and Use Of Examples
By Algebraists: An Exploratory Study John Paul Cook
& Tim FukawaConnelly This paper reports on an
exploratory study of 10 algebraists designed to investigate the reasoning
behind their selection of examples for their own teaching and research.
Variation theory provided a lens with which to analyze the algebraists goals
for their collections of examples and to speculate about the resulting
pedagogical implications. Though findings from this exploratory study should
be regarded only as preliminary and in need of further justification, our
results provide some initial evidence that mathematicians use a relatively
small number of very wellchosen classes of examples in both their teaching
and their research (suggesting that this might be a useful pedagogical
strategy for students as well). We also report on the examples of groups and
rings that the algebraists deemed to be the most important for students of
introductory abstract algebra. 
9:25 –
9:55 am 
Session 7 – Preliminary Reports

Wind Star 
Student Calculus Reasoning Contexts Matthew Petersen, Sarah Enoch & Jennifer Noll This paper analyzes how student discourse about Calculus is situated
in a graphical representation of a physics problem. Students were asked to
identify three unlabeled graphs as representing the position, velocity and
acceleration of a car. Findings showed that the students reasoned in three
distinct contexts  staticgraphical, covariational, and physical. While the
students were able to communicate effectively between the first two contexts,
and leverage them to find a solution to the problem, the students' discourse
in the physical context did not communicate well with their discourse in the
other two contexts, nor was it very fruitful in finding a solution to the
problem. 
Grand Mesa B 
Preservice Teachers Uses Of The
Internet To Investigate The Proof Of The Pythagorean Theorem and Its Converse Aaron Brakoniecki Learners of mathematics,
including preservice teachers, often explore online resources when
investigating mathematical problems. When asked to search online for
resources that would help them be able to better explain a proof of the
Pythagorean theorem and its converse, preservice teachers used a variety of
different searching strategies to locate information. Further, the ways in
which this information was incorporated into their understanding of
mathematics became evident through concept maps. This proposal describes the
study conducted and initial results from the data and asks the reader to
consider possible ways this research might be extended and refined. 
Grand Mesa C 
Developing PreService Secondary
Math Teachers Capacity With Error Analysis Related To MiddleGrades
Mathematics Diana White As part of a National Science
Foundation Noyce Scholarship Grant, one university substantially revised its
preservice secondary (grades 712) math teacher preparation program. As one
component of this program, preservice teachers take three credit hours of
middle level number and operation and geometry, with a focus on mathematical
knowledge for teaching. As a research component, we investigated the impact
of this course on preservice teachers capacity to identify, analyze, and
respond to student errors. This paper provides additional background and
results from the first two offerings of the course, as well as ideas for
further study. 
Chasm Creek A 
Investigating Instructors Concerns
About Assessments In InquiryBased Learning Methods Courses Inah Ko & Vilma Mesa We present initial findings of ongoing research that investigates the
nature of instructors concerns as they design and use assessments for their
students using inquirybased learning (IBL) approaches. Using data collected
from biweekly onlineteaching logs written by 39 instructors, we categorized
concerns into three major themes: Item Design/Assessment, Course/Resources,
and Student difficulty. We compare two areas of concerns (designing
assessment and using quizzes, tests, and exams) according to the type of
concern and the instructors experience with IBL, course level, and year by
using the frequencies of each category cited for each log. Our work will
contribute to IBL research by analyzing instructors challenges as a
preliminary study to enhancing IBL teaching and learning in college
mathematics education. 
Chasm Creek B 
Teaching InquiryBased Mathematics
To InService Teachers: Results From The Field Karen Keene & Celethia McNeil We present results from a classroom teaching data collection that
involved practicing teachers as they participated in an inquiryoriented
differential equations course. Data was collected to investigate how the
teachers participation in this kind of course, different from any of their
original mathematics courses, may influence their conceptions of teaching,
mathematics, and student learning. Preliminary results indicate that teachers
were changed by their experience in the class, at least as expressed in
interviews. The teachers were likely to attempt to use more studentcentered
methods in their classroom and believe that student learning is better in the
studentcentered environment. Additionally, attitudes about nonlecture,
although mixed, did indicate more positive attitudes towards the
constructivist perspective on learning. Finally, the teachers participation
in argumentation increased during the course. 
9:55 –
10:25 am Atrium 
Coffee Break

10:25
– 10:55 am 
Session 8 – Contributed Reports 
Chasm Creek B 
Graduate Students Teaching
Assistants (GTAs) Beliefs, Instructional Practices, and Student Success Jessica Ellis In this report I present findings from a large, national study focused
on Calculus I instruction. Graduate student Teaching Assistants (GTAs)
contribute to Calculus I instruction in two ways: : as the primary teacher
and as recitation leaders. As teachers, GTAs are completely in charge of the
course just as a lecturer or tenured track/ tenured faculty would be,
although they lack the experience, education, or time commitment of their
faculty counterparts. In this study, I investigate how GTAs compare to tenure
track/tenured faculty, and other full/part time faculty on their (a) beliefs
about mathematics; (b) instructional practices; and (c) students success in
Calculus I. Findings from this report point clearly to a need to prepare GTAs
adequately for the teaching of calculus but also for further examination of
the nature and implications of the differences between GTA and other instructor
types beliefs about teaching and teaching practices. 
Grand Mesa A 
Undergraduate Students Stochastic
Understanding Of Probability Distribution Darcy Conant Stochastic conceptions undergird development of conceptual connections
between probability and statistics and support development of a principled
understanding (Greeno, 1978) of probability distribution. This study employed
mixed research methods to investigate the impact of an instructional course
intervention designed to support development of stochastic understanding of
probability distribution. Instructional supports consisted of supplemental
lab assignments comprised of anticipatory tasks designed to engage students
in coordinating thinking about complementary probabilistic and statistical
notions along a hypothetical learning trajectory aimed at development of
stochastic understanding of probability distribution. Participants were 184
undergraduate students enrolled in a lecture/recitation, calculusbased,
introductory probability and statistics course. Results of quantitative
analyses showed completion of stochastic lab assignments had a statistically
significant impact on students stochastic understanding of probability
distribution. Student interviews revealed those who held stochastic
conceptions also indicted integrated reasoning related to probability,
variability, and distribution and presented images supporting principled
understanding of probability distribution. 
Grand Mesa B 
Technology and Algebra In Secondary
Mathematics Teacher Preparation Programs Eryn Stehr & Lynette Guzman Most recently, the Conference Board of the Mathematical Sciences has
advocated for incorporating technology in secondary mathematics classrooms.
Colleges and universities across the United States are incorporating
technology to varying degrees into their mathematics teacher preparation
programs. This study examines preservice secondary mathematics teachers
opportunities to expand their knowledge of algebra through the use of
technology and to learn how to incorporate technology when teaching algebra
in mathematics classrooms. We explore the research question: What
opportunities do secondary mathematics teacher preparation programs provide
for PSTs to encounter technologies in learning algebra and learning to teach
algebra? We examine data collected from a pilot study of three Midwestern
teacher education programs conducted by a larger project investigating
algebra. Our data suggest that not all secondary mathematics teacher
preparation programs integrate experiences with technology across mathematics
courses, and that mathematics courses may provide few experiences with
technology to PSTs beyond strictly computational. 
Chasm Creek A 
Understanding Students' Conceptualizations
Of Logical Tools Casey Hawthorne While a significant amount of research has been devoted to exploring
why university students struggle applying logic, limited work can be found on
how students actually make sense of formal logic itself and the logical
mechanisms used to communicate logical equivalence. This project borrows the
theoretical framework of unitizing and reification, that have been
effectively used to explain the types of integrated understanding required to
make sense of symbols involved in numerical computation and algebraic
manipulation, to investigate students conceptualization of truth tables and
the implication statements. By using a continuum as a framework to analyze
the degree to which students thinking of each is compartmentalized versus
unified, results indicate that students tend to favor one logical mechanism
over another, without establishing a holistic view of both or an integrated
view of the two together. 
11:05
– 11:35 am 
Session 9 – Contributed Reports

Chasm Creek B 
Comparing Calculus Students
Representation Use Across Different Settings Dov Zazkis The distinction between analytic (notationbased) and visual
(diagrambased) representations within students mathematical problemsolving
has been part of the mathematics education literature for more than 40 years.
However, in spite of this long history there are many unanswered questions
regarding how and why particular students choose particular representations,
and what influences their social surroundings have on their individual
representation use. This study coordinates analyses of undergraduate calculus
students analytic and visual reasoning across both oneonone interview and
groupwork settings. These analyses help clarify differences between
individual representation use and representation use in group settings. 
Chasm Creek A 
An Investigation Of College
Students Statistical Literacy Erin Glover & Sean Larsen Many statistics educators consider statistical literacy a vital skill because
it supports students in thinking critically about the way data is used every
day in social, political, and medical contexts. An important component of
statistical literacy is the ability to read, interpret, and contextualize
graphical information. As part of a classroom teaching experiment in an
introductory college statistics course, students were given a set of graphs
and asked to interpret the graphs, compare them and to describe real life
contexts that might explain differences in the graphs. This research
presentation will share an analysis of student responses to this task, with a
particular focus on students use of statistical language and their abilities
to contextualize situations that would produce the given data. Implications
for future research as well as pedagogical implications will be discussed. 
Grand Mesa A 
What Is Simplifying?: Using Word
Clouds As A Research Tool Benjamin Wescoatt This paper describes the utilization of word clouds within a research
methodology. To explore student notions of the concept of simplify in a
trigonometry course, students responded to the prompt In your own words,
what does it mean to simplify? The researcher created a word cloud derived
from the student responses to explore and identify themes. These themes
formed an initial framework for an indepth analysis of the responses. During
the textual analysis, the word cloud was consulted to confirm findings. Using
the word cloud in preliminary and confirmatory roles adhered to the framework
put forth by McNaught and Lam (2010). From the analysis, students appeared to
view the act of simplifying as a process of taking an expression to its most
basic state in order to reduce the perceived size (physical or cognitive) of
the expression. Moreover, word clouds played a valuable role, providing
visual representations of data. 
Grand Mesa B 
Prospective Secondary Teachers
Conceptions Of Proof and Interpretations Of Arguments Annamarie Conner, Richard Francisco, Carlos Nicolas Gomez,
Ashley Suominen & Hyejin Park We analyzed the interviews of three prospective secondary mathematics
teachers to examine their conceptions of proof and how they validated
arguments in the context of students answers. Our participants had differing
views of the definition of proof and its role in mathematics. Their work when
validating arguments in large part aligned with their professed views of
proof, with some deviations on the part of one participant. Further research
must examine whether this consistency is prevalent across prospective
teachers and how this relates to teachers work with proof in classrooms. 
11:40
– 12:40 pm Grand Mesa DEF 
Lunch

12:45
– 1:15 pm 
Session 10 – Preliminary Reports

Grand Mesa C 
Student Understanding Of The
Fundamental Theorem Of Calculus At The MathematicsPhysics Interface Rabindra Bajracharya & John Thompson We studied students understanding of the Fundamental Theorem of
Calculus (FTC) in graphical representations that are relevant in physics
contexts. Two versions of written surveys, one in mathematics and one in
physics, were administered in multivariable calculus and introductory
calculusbased physics classes, respectively. Individual interviews were
conducted with students from the survey population. A series of FTCbased
physics questions were asked during the interviews. The written and interview
data have yielded evidence of several student difficulties in interpreting or
applying the FTC to the problems given, including attempting to evaluate the
antiderivative at individual points and using the slope rather than the area
to determine the integral. The interview results further suggest that
students often fail to make meaningful connections between individual
elements of the FTC. 
Grand Mesa B 
Transforming Remedial Mathematics
Instruction With HighQuality Peer Teaching Kristen Bieda, Raven Mccrory & Steven Wolf This project investigated the potential of a hybrid remedial
mathematics course (RMC), taught by a corps of undergraduate peers in a secondary
mathematics teacher preparation program, to provide remedial mathematics
students with opportunities to develop robust mathematical proficiency. We
collected data from four semester exams as well as a final exam for students
in both intervention and control sections of the RMC. We also conducted
interviews with students in the intervention section of the RMC as well as
prospective secondary mathematics teachers (PSMTs) who served as instructors
for the course. Our findings show that the instructional model appears to
positively impact the learning of students in a RMC, while also providing
PSMTs with meaningful opportunities to learn to teach. We will share
revisions to our instructional model and seek audience input about approaches
to scaling this model to remedial mathematics courses at other institutions. 
Wind Star 
Mathematicians Views On
TransitionToProof and Advanced Mathematics Courses Milos Savic, Melissa Mills, & Robert Moore This study explores mathematicians views on 1) knowledge and skills
students need in order to succeed in subsequent mathematics courses, 2)
content courses as transitiontoproof courses, and 3) differences in the
proving process across mathematical content areas. Seven mathematicians from
three different universities (varying in geographic location and department
size), were interviewed. Precision, sensemaking, flexibility, definition
use, reading and validating proofs, and proof techniques are skills that the
mathematicians stated were necessary to be successful in advanced mathematics
courses. The participants agreed unanimously that a content course could be
used as a transitiontoproof course under certain conditions. They also
noted differences in the proving processes between abstract algebra and real
analysis. Results from this study will be used to frame a larger study
investigating students proof processes in their subsequent mathematics
content courses and investigating how these skills can be incorporated into a
transitiontoproof course. 
Chasm Creek B 
Current and Future Faculty Members
Mathematical Knowledge For Teaching Calculus Natasha Speer & Shahram Shawn Firouzian Findings from research into mathematical knowledge for teaching have
informed the design of preparation and professional development programs for
K12 teachers. At the college level there has been limited research into
mathematical knowledge for teaching. We lack findings that demonstrate that
expert teachers of college mathematics know and make use of knowledge beyond
solely mathematical content.. The goal of this study is to examine the
knowledge of student thinking possessed by mathematicians who teach calculus.
Data come from interviews on student thinking about core calculus concepts..
Interviewees were research mathematicians who have been recognized for their
teaching excellence and mathematics graduate students.. Findings demonstrate
that the mathematicians were more able to identify known student difficulties
as well as to describe common strategies students use to successfully solve
the problems. Implications for research and professional development for
novice college mathematics instructors are discussed. 
Chasm Creek A 
Assessment In Undergraduate InquiryBased
Learning Mathematics Courses Timothy Whittemore & Vilma Mesa We report initial findings of a study that seeks to investigate the
methods instructors use to assess their students learning and how these
assessments affect the instruction in their classrooms. Using data collected
from 13 instructors using inquirybased learning methods, we seek to discuss
the instructors goals for the students, the ways they measure the students
progress towards these goals, the feedback they give students, and how these
assessments affected their instruction. Our analysis of the data uses open
coding of the transcripts and of the documents (e.g., syllabi, exams,
homework assignments) that the instructors gave to the students. Instructors
cite using informal assessments and focusing on presentations when asked
about knowing that students are learning. They cite formal assessments and
examinations when asked about measuring that students are learning. We seek
input on the analysis of the materials as current results may depend on the
coding system used. 
Grand Mesa A 
Professional Development and
Student Achievement On Standardized State Exams Melissa Goss, Rebecca Anne Dibbs & Robert Powers Although teacher quality is positively correlated with student
achievement, easily quantified measures of teacher quality are not accurate
measures of quality; teacher pedagogical content knowledge and skills are
better predictors, but difficult to measure. Professional development may be
a costeffective vehicle for developing new skills in inservice teachers,
but there is conflicting research on whether professional development
measurably raises student achievement on high stakes standardized tests. The
purpose of this causalcomparative study was to examine Andrew, an
inservice, high school teacher participant in the masters program. State
mathematics assessment and student demographic data were collected from
school districts for 4 academic years spanning from preprogram through
program completion. Oneway ANOVA analysis on student scale scores factoring
by year showed a significant decrease in student mathematics scale scores
potentially attributable to differences in population. Independentsamples t
tests on the final two years showed a statistically insignificant increase in
student growth percentiles. 
1:25 –
1:55 pm 
Session 11 – Theoretical Reports

Chasm Creek B 
The Duality Principle and Learning
Trajectories In Mathematics Education Eric Weber & Elise Lockwood The purpose of this paper is to argue that attention to students ways
of thinking should complement a focus on students understanding of specific
mathematical content, and that attention to these issues can be leveraged to
model the development of mathematical knowledge over time using learning
trajectories. To illustrate the importance of ways of thinking, we draw on
Harels (2008a, 2008b) description of mathematical knowledge as comprised of
ways of thinking and ways of understanding. We use data to illustrate the
explanatory and descriptive power that attention to the duality of ways of
understanding and ways of thinking provides, and we propose suggestions for
constructing learning trajectories in mathematics education research. 
Chasm Creek A 
What Is A Proof? A Linguistic
Answer To A Pedagogical Question Keith Weber Proof is a central concept in mathematics education, yet
mathematics educators have failed to reach a consensus on how proof should be
conceptualized. I advocate defining proof as a clustered concept, in the
sense of Lakoff (1987). I contend that this offers a better account of
mathematicians practice with respect to proof than previous accounts that
attempted to define a proof as an argument possessing an essential property,
such as being convincing or deductive. I also argue that it leads to useful
pedagogical consequences. 
Grand Mesa A 
The Construction Of Cohomology As
Objectified Action Anderson Norton The purpose of this paper is to
investigate a theory about the nature of mathematical development, in which
mathematics is characterized as the objectification of action. Informed by
existing research on how students construct new mathematical objects, we
consider as an example, the psychological construction of cohomology and
related objects of algebraic topology. This example is used to test theories
of mathematical development through extension and through the authors lived
experience. Findings from the selfstudy are used to integrate existing
research on students learning of abstract algebra, particularly from a
neoPiagetian perspective. 
2:05 –
2:35 pm 
Session 12 – Theoretical Reports 
Grand Mesa A 
An Origin Of Prescriptions For Our Mathematical
Reasoning Yusuke Uegatani To build a supplementary theory
from which we can derive a practical way of fostering inquiring minds in
mathematics, this paper proposes a theoretical perspective that is compatible
with existing ideas in mathematics education (radical constructivism, social
constructivism, APOS theory, David Talls framework, the framework of
embodied cognition, new materialist ontologies). We focus on the fact that
descriptive and prescriptive statements can be treated simultaneously, and
consider both descriptive and exemplary models in our minds. This indicates
that descriptive statements in mathematics come from our descriptions of
models, and prescriptive statements come from the exemplarity of exemplary
models. As a practical suggestion from the proposed perspective, we point out
that careful communication is needed so that inquiring minds do not recognize
the refutation of their arguments as a denial of their way of mathematical
thinking. 
Chasm Creek B 
Disambiguating Research On Logic As
It Pertains To Advanced Mathematical Practice Paul Dawkins Many consider logic a hallmark of mathematical practice and an
integral part of prooforiented mathematical instruction. This is true of the
term "logic" whether it refers to a domain of mathematical study or
to aspects of reasoning, but I claim that these formalized and psychological
senses of the term must be carefully distinguished in mathematics education
research. In the course of identifying how the abstraction criterion has been
misapplied across various types of logic in psychological and mathematics
education research, I outline a framework for the disambiguation of the range
of research constructs referred to as "logic". By distinguishing
the types of logic pertinent to mathematics education instruction, I hope to
provide a language by which future research can better specify the constructs
they investigate. Clearer research constructs should help the community to understand
the role various logics play in students apprenticeship into the practices
of advanced mathematics. 
Chasm Creek A 
Two Metaphors For Realistic Mathematics
Education Design Heuristics: Implications For Documenting Student Learning Estrella Johnson The primary goal of this work
is to articulate a theoretical foundation based on Realistic Mathematics
Education (RME) that can support the analysis of student learning. To do so,
I will first frame the guided reinvention and emergent models design
heuristics separately in terms of both increasingly general student activity
and in terms of concept development. Then, I will consider how the RME design
heuristics could inform how one conceptualizes student learning. To do so, I
will draw on two metaphors for learning and, by drawing on these two
perspectives, propose ways in which the RME design heuristics can inform the
analysis of student learning. 
2:35 –
3:05 pm Atrium 
Coffee Break 
3:05 –
3:35 pm 
Session 13 – Preliminary Reports 
Chasm Creek A 
Characterizing Mathematical
Complexity Of Tasks In Calculus I Nina White, Vilma
Mesa & Cameron Blum We present findings from a
revised framework created to analyze tasks that calculus teachers assign
their students. In the presentation we will highlight the features of the
analytical framework and the steps taken to ensure high intercoder
reliability. The framework has been used to analyze all tasks (N=2,996)
present in homework, quizzes, and exams from six faculty teaching Calculus I
in two twoyear colleges. We highlight some insights we have gained in
creating this framework and possible uses by other researchers and other
contexts. 
Chasm Creek B 
The Value Of Systematic Listing In
Correctly Solving Counting Problems Elise Lockwood &
Bryan Gibson Although counting problems are
easy to state and provide rich, accessible problem solving situations, there
is much evidence that students struggle with solving counting problems
correctly. With combinatorics (and the study of counting problems) becoming
increasingly prevalent in K12 and undergraduate curricula, there is a need
for researchers to identify potentially effective instructional interventions
that might give students greater success as they solve counting problems. In
this study, we tested one such intervention – having students engage in
systematic listing of what they were trying to count. We found that even
creating partial lists of the set of outcomes was a significant factor in
students success on problems. Our findings suggest that more needs to be
done to refine instructional interventions that will facilitate listing. We
discuss these findings, suggest followup studies, and request feedback from
the audience. 
Grand Mesa A 
Student Conceptions Of
Trigonometric Identities Through Apos Theory Benjamin Wescoatt This preliminary study attempts
to describe an initial genetic decomposition of a trigonometric identity for
college students. Scant research exists into the concepts found in
trigonometry. Thus, little is known about how students actually understand a
trigonometric identity. Following the guidelines of APOS theory, an initial
genetic decomposition for a trigonometric identity was proposed. According to
this decomposition, students with action conceptions can verify identities
explicitly using stepbystep manipulations while students holding a process
conception are able to visualize steps to demonstrate that the identity is
true. Having an object conception means students recognize the truth of the
equality without verification and are able to then use the identity to verify
other identities. After observing students in taskbased interviews, needed
modifications to the genetic decomposition became apparent. For example,
students conceptions of the function argument appeared to influence the
verification process. 
Grand Mesa B 
What Constitutes A WellWritten
Proof? Robert Moore The purpose of this study was
to identify some of the characteristics mathematicians value in good proof
writing. Four mathematicians were interviewed. First, they evaluated and
scored six proofs of elementary theorems written by students in a discrete
mathematics or geometry course, and second, they responded to questions about
the characteristics they value in a wellwritten proof and how they
communicate these characteristics to students. Preliminary results indicate
that these mathematicians agreed that the most important characteristics of a
wellwritten proof are (a) correct logic and (b) clarity. Although these
mathematicians differed in the attention they give to layout, grammar,
punctuation, and mathematical notation, they agreed in giving these
characteristics relatively little weight in the overall score. The results
also showed that, in addition to demonstrating good proof writing in class,
writing comments on students papers is an important way they teach their
students to write good proofs. 
Wind Star (1^{st} Floor) 
Characteristics Of Successful
Programs In College Calculus: How Calculus Instructors Talk About Their
Students Sean Larsen,
Estrella Johnson & Dov Zazkis The CSPCC (Characteristics of
Successful Programs in College Calculus) project is a large empirical study,
investigating mainstream Calculus 1, that aims to identify the factors that
contribute to successful programs. The CSPCC project consists of two phases. Phase
1 entailed largescale surveys of a stratified random sample of college
Calculus 1 classes across the United States. Phase 2 involves explanatory
case study research into programs that were identified as successful based in
part on the results of the Phase 1 survey. During our case study site visits,
we interviewed calculus instructors and asked a number of questions that
prompted them to discuss their students. The purpose of the analyses we will
present here is to characterize the ways that calculus instructors talk about
their students. To do so, we will examine instructor survey responses and
analyze instructor interviews conducted at the casestudy institutions (PhD
and Bachelors granting levels). 
Grand Mesa C 
Student Views About Truth In
Axiomatic Mathematics Brian Katz An undergraduate mathematics
major should come to hold appropriate views about the conclusions reached by
our disciplinary methods. This project explores the views about truth in
axiomatic mathematics of a group of students who are (mostly) in their final
proofbased course, Modern Geometry. Do these students hold expertlike views
about truth in mathematics, and do those views change during a course that
emphasizes epistemological themes? I find preliminarily that many of these
experienced students do not distinguish the truthvalue of theorems from that
of definitions or axioms at the start of the term, but they develop more expertlike
perspectives on truth during the course. 
3:45 – 4:15 pm 
Session 14 – Contributed Reports 
Grand Mesa B 
ModelOf To ModelFor In The
Context Of Riemann Sum Kritika Chhetri
& Jason Martin This research focuses on mental
challenges that students face and how they resolve these challenges while
transitioning from intuitive reasoning to constructing a more formal
mathematical structure of Riemann sum while modeling real life contexts. A
pair of Calculus I students who had just received instruction on definite
integral defined using Riemann sums and illustrated as area participated in
ten interviews. They were given three contextual problems related to Riemann
sums but were not informed of this relationship. While modeling these problem
situations, the intent was to observe students transitioning from modelof
to modelfor reasoning based on Gravemeijer and Stephan (2010). Results
indicate that it was not the end results but records of their ways of acting
and reasoning about their contextual problem through multiple representations
along with real life intervention that served as tools for supporting their
transition from modelof informal activities to modelfor more formal
mathematical reasoning. 
Chasm Creek B 
Student Understanding Of Mean,
Distribution and Standard Deviation Samuel Cook &
Tim FukawaConnelly This study investigates the
understandings of mean, median, distribution and standard deviation that
undergraduate students have at the end of an introductory statistics course.
The goal was to explore their understandings as a followup to previous
studies documenting incoming student difficulties with the concepts and
determine whether a course would help them achieve a more statistically
appropriate understanding. They overwhelmingly think about the mean as the
average and via the calculating formula, meaning they understand it as a
process. Similarly, they understand the median in terms of the process for
determining it, or via the locationbased term, middle. As a result,
students do not generally understand the two measures to be describing a
similar concept. Students do, reliably connect the shape of a distribution to
standard deviation, but that connection varies by type of display and is not
based on a reliable rule. 
Grand Mesa A 
Preservice Secondary Teachers'
Understanding Of The Cartesian Connection and Equivalence Kyunghee Moon Both prior research and
national standards emphasize the importance of critical ideas, such as the
Cartesian Connection and equivalence, in algebra problem solving. The
mathematics education community, however, has yet to determine whether the
secondary teachers who teach such ideas fully grasp these ideas themselves.
To investigate this, I interviewed a cohort of nine preservice secondary
teachers in a teacher education program with two algebra problems that embed
these ideas. The results showed that many of the teachers failed to
understand equivalence as a relation between geometric objects, and thus
could not solve an algebra problem by relating algebraic equations to their
corresponding graphs. Many also misinterpreted the meaning of the term
solution, and thus could not use the Cartesian Connection to find a
solution of an equation. It is advisable that secondary teacher education
programs focus more on these critical ideas so that secondary teachers can
impart such ideas on their students. 
Chasm Creek A 
Living It Up In The Formal World:
An Abstract Algebraists Teaching Journey John Paul Cook,
Ameya Pitale, Ralf Schmidt & Sepideh Stewart Abstract algebra is a
fascinating field of study among mathematics topics. Despite its importance,
very little research has focused on the teaching of abstract algebra. In
response to this deficiency, in this study we present an abstract algebra
professors daily activities and thought processes as shared through his
teaching diaries with a team of two mathematics educators and another
abstract algebraist over the period of two semesters. We examined how he was
able to live in the formal world of mathematical thinking while also dealing
with the many pedagogical challenges that were set before him during the
lectures. 
4:25 –
4:55 pm 
Session 15 – Contributed Reports 
Chasm Creek A 
Are Students Better At Validation
After A TransitionToProof Course? Annie Selden &
John Selden This paper presents the results
of an empirical study of the proof validation behaviors of sixteen
undergraduates after taking a transitiontoproof course that emphasized
proof construction. Students were interviewed individually towards the end of
the course using the same protocol used by Selden and Selden (2003) at the
beginning of a similar course. Results include a description of the students
observed validation behaviors, a description of their proffered evaluative
comments, and the suggestion that taking a transitiontoproof course does
not seem to enhance students validation abilities. We also discuss
distinctions between proof validation, proof comprehension, proof
construction and proof evaluation and point out the need for future research
on how these concepts are related. 
Grand Mesa A 
Considering Mathematical Practices
In Engineering Contexts Focusing On Signal Analysis Reinhard Hochmuth,
Rolf Biehler & Stephan Schreiber In the light of a rough description of the different contexts in which
mathematics is learned and used in engineering studies, this report addresses
epistemic relations between mathematics in higher mathematics lectures and
mathematics in advanced engineering courses. In particular it elaborates on
how different meanings of symbols, as subjectively relevant aspects of
mathematical objects, are related to different institutional contexts and
their dominant discourses. It is argued that modeling cycles are not an
adequate tool in this context. Instead, we suggest using concepts from
Anthropological Theory of Didactics (ATD). Inspired by (Castela & Romo
Vzquez, 2011), exemplarily concepts from ATD are applied to topics and data
from signal analysis. Finally, we claim this research could serve as a step
towards investigating empirical questions relevant to students learning and
competences and, in particular, optimizing curricula and teaching in
undergraduate mathematics. 
Chasm Creek B 
Evaluating Professional Development
Workshops Quickly and Effectively Charles Hayward &
Sandra Laursen Many funding agencies require
evaluation of the impact of professional development projects they support.
However, improved student outcomes, the ultimate goal, may take longer to be
realized than the project time frame allows. Instructors need time to
implement and refine new skills before positive student outcomes are
realized, a delay that may be exacerbated in classes that are not taught
frequently. We report on an efficient and costeffective selfreport measure
designed to detect the initial changes in teaching practices that lead to
improved student outcomes over time. We discuss the ability for timely and
accurate measures through this instrument. Results support the interpretation
that instructors reported teaching practices show changes consistent with
methods taught at professional development workshops on InquiryBased
Learning in mathematics. Additionally, correlations with selfreported level
of implementation suggest that instructors are reporting honestly, and not
just socially desirable changes consistent with their concept of real
InquiryBased Learning. 
Grand Mesa B 
An Eye To The Horizon: The Case Of
Delias Hexagon Ami Mamolo This paper explores preservice
secondary school mathematics teachers preferences when advising a student on
how to determine the area of an irregular hexagon. The research attends to
participants personal mathematical knowledge, as interpreted through the
lens of Knowledge at the Mathematical Horizon. Philosophical notions of inner
and outer horizons of conceptual objects are adapted to provide a refined
analysis of participants personal strategies and preferences as evoked by an
unconventional problem. The interplay amongst participants understanding of
mathematical structure, their focus of attention when interpreting a problem,
and the advice they offer to a student are of interest. Implications for
teacher education and further avenues of research are suggested. 
5:10 –
6:10 pm Grand Mesa DEF 
Plenary SessionPlenary
Speaker: Anna Sfard Mathematics
Learning: Does Language Make A Difference? Abstract: Mathematics and its
learning are generally believed to be relatively independent of the language
in which they are practiced. This assumption tacitly underlies the nowadays
popular idea of international comparisons such as TIMSS or PISA, in which
young people from all over the world are being tested with the help of a
single mathematical questionnaire. The fact that the questions appear in
different languages does not diminish examiners conviction that, wherever
they go, they are testing the same mathematics, thus assessing fully
comparable types of learning. And yet, in the view of recent theoretical
developments and some new empirical findings, the assumption about the
languageproof nature of mathematics and its learning may be questioned. This
issue is of particular importance to those who teach mathematics in schools
and universities. Indeed, if it turns out that the way people learn is shaped
by their main language, there may be significant differences in the needs of
learners gathered in the same multilingual classroom. The question of the impact of language on
mathematics learning is the focus of this talk. I will begin with a brief
historical survey of research guided by the famous SapirWhorf Hypothesis,
according to which all human thinking is shaped by language. I will follow
with a theoretical reflection on the relation between thinking and
communication, undertaken in an attempt to reconceptualize the topic. I will then
use the resulting conceptual apparatus while summarizing and interpreting
results of two studies, one on learning limits and infinity and the other on
learning fractions and probability, both of them launched in the quest for
dissimilarities in mathematical discourses of learners coming from different
linguistic backgrounds. 

Dinner On Your Own 
Saturday, March 1, 2014
9:00 –
9:30 am 
Session 16 – Contributed Reports

Chasm Creek B 
Students Struggle With The
Temporal Order Of Delta and Epsilon Within The Formal Definition Of A Limit Aditya Adiredja &
Kendrice James Studies about students
understanding of the formal definition of a limit, or the epsilon delta
definition suggest that the temporal order of delta and epsilon is an
obstacle in learning the formal definition. While such difficulty has been
widely documented, patterns of students reasoning are largely unknown. This
study investigates the degree of difficulty students have with the temporal
order, along with justifications that students provide to support their
claim. diSessas Knowledge in Pieces provides a suitable framework to explore
the context specificity of students knowledge as well as the potential
productivity of their prior knowledge in learning. 
Chasm Creek A 
The Construction Of A Video Coding
Protocol To Analyze Interactive Instruction In Calculus and Connections With
Conceptual Gains Matthew Thomas Instruments called concept
inventories are being used to investigate students' conceptual knowledge of
topics in STEM fields, including calculus. One interactive instructional
style called InteractiveEngagement has been shown to improve students' gains
on such instruments in physics. In this paper, we discuss the development of
a video coding protocol which was used to analyze the level of
InteractiveEngagement in calculus classes and investigate the correlation
with gains on the Calculus Concept Inventory. 
Grand Mesa A 
Academic and Social Integration
Revealed In Characteristics Of Successful Programs In College Calculus
Project: The TwoYear College Context Vilma Mesa, Nina
White & Helen Burn We present an analysis of
features common across four Calculus I programs at twoyear colleges
identified as successful in the Characteristics of Successful Programs in
College Calculus (CSPCC) study. In this paper we discuss how these features
emerged in the analysis of the four cases and their connection to theories of
student academic and social integration. Student academic and social
integration have been identified as closely related to student persistence in
college. We used a constant comparative analysis to identify themes within
and across institutions, using transcripts of 22 interviews with faculty,
staff, and administrators, and student focus groups. We discuss three of the
seven major themes that arose, High quality instructors, Faculty autonomy and
trust in the teaching of calculus, Supporting students academically and
socially, and Attention to placement, which support a model of student
academic and social integration. We present further research steps and some
implications for practice. 
Grand Mesa B 
Teaching The Concept Of
Mathematical Definition Using Student Construction and SelfAssessment Susanna Molitoris
Miller Definitions are an important
part of the study of mathematics, yet many students struggle with
successfully understanding and using this construct. It has been suggested
that students may improve their understanding of mathematical definitions by
engaging in the act of writing definitions (de Villiers, Govender, &
Patterson, 2009). Through a mixture of survey and teaching experiment
methodology this study explores preservice elementary teachers
understanding of mathematical definitions before and after engaging in a
teaching experiment which provided many opportunities for the participants to
write their own mathematical definitions for familiar and novel classes of
quadrilaterals. Definitions were assessed as having necessary, sufficient and
minimal conditions. It was found that while many students initially struggled
to write definitions that meet these qualifications, the process of trying to
construct their own definitions did improve students understanding of these
characteristics of mathematical definitions. 
9:40 –
10:10 am 
Session 17 – Preliminary Reports

Grand Mesa B 
Differential Participation In
Formative Assessment and Achievement In Undergraduate Calculus Rebecca Dibbs &
Michael Oehrtman Prior formative assessment
research has shown positive achievement gains when classes using formative
assessment are compared to classes that do not. However, little is known
about what, if any, benefits students that are not participating regularly in
formative assessment gain from these assignments. The purpose of this study
was to investigate the achievement of the students in two introductory
calculus courses using formative assessment at the three different
participation levels observed in class. Although there was no significant
difference on any demographic variable other than gender and no significant
difference in any achievement predictive variables between the groups of students
at the different participation levels, there were significant differences in
achievement on all but the first activity writeup and the final exam. 
Grand Mesa A 
Cognitive Processes and Knowledge
In Activities In Community College Trigonometry Lessons Linda Leckrone &
Vilma Mesa Over 50,000 students take
trigonometry at twoyear colleges in the U.S., yet little is known about
their instruction. We report an analysis of activities in trigonometry
classes taught at a community college attending to two dimensions, the type
of knowledge used (Factual, Procedural, Conceptual, and Metacognitive) and
the cognitive processes (Remember, Understand, Apply, Analyze, Evaluate,
Create) intended in the activity as enacted by teachers in their lessons.
Most of the 163 activities were classified as applying procedural knowledge;
over onefifth of the activities were coded as remembering factual knowledge
or understanding conceptual knowledge. We discuss these findings in light of
the community college setting and offer some questions for further research. 
Chasm Creek B 
Using The Flipped Model To Address
Cognitive Obstacles In Differential Equations Jenna Tague,
Jennifer Czocher, Amanda Roble & Gregory Baker Recent work has shown that
there is a lack of coherence from calculus to differential equations: the
knowledge calculus knowledge students are expected to gain by the end of the
calculus sequence is different from how that knowledge is expected to be used
in differential equations (Authors). In this report, we describe how we have
begun to address some of these issues with coherence through utilizing the
flipped classroom model. We share our theoretical perspective, how it was
enacted using the classroom model and technology, and also a preliminary
evaluation of students expectations of and perceptions of the coherence of
the course and its content. 
Chasm Creek A 
An Exploration Of Mathematics
Graduate Teaching Assistants Teaching Philosophies Kedar Nepal This is an investigation of the teaching philosophies of beginning
mathematics graduate teaching assistants. Three teaching philosophy
statements from each of four participants were collected at different stages
of a semesterlong teaching assistant preparation program and analyzed.
Principal elements found in these statements before they underwent training
and how their philosophies changed over time during training will be
discussed. 
Wind Star 
Approximation: A Connecting
Construct Of The FirstYear Calculus? Kimberly Sofronas, Thomas Defranco, Hariharan Swaminathan,
Charles Vinsonhaler, Nicholas Gorgievski & Brianna Wiseman This report will present preliminary findings from a research study
designed to investigate calculus instructors perceptions of approximation as
a central concept and possible unifying theme of the firstyear calculus. The
study will also examine the role approximation plays in participants
selfreported instructional practices. A survey was administered through
Qualtrics to a stratified random sample of 3930 mathematicians at higher
education institutions throughout the United States with a desired N = 300.
Quantitative and qualitative methods were used to analyze the data gathered.
Findings from this research will contribute to what is known about the
perceptions and teaching practices of calculus instructors regarding the role
of approximation in firstyear calculus courses. Researchbased findings
related to the role of the approximation concept in the firstyear calculus
could have implications for firstyear calculus curricula. 
Grand Mesa C 
Noticing The Math In Issues Of
Social Justice Ami Mamolo This preliminary report examines preservice secondary mathematics
teachers engagement with problems which contextualized mathematics in issues
of social justice. A framework for Teaching Mathematics for Social Justice
was employed and participant responses were analysed with respect to what
mathematics they noticed and attended to in and after the problem solving.
Results suggest participants had difficulty seeing the math in nonmath
contexts, and that their ability to notice the embedded mathematics was
influenced by the specific social context as well as their orientation
towards mathematics (both in general and regarding specific content).
Implications for research and teacher education are described. 
10:10
– 10:40 am Atrium 
Coffee Break 
10:40
– 11:10 am 
Session 18 – Preliminary Reports

Chasm Creek A 
Presentation Of Matrix
Multiplication In Introductory Linear Algebra Textbooks John Paul Cook & Sepideh Stewart We conducted an analysis of 17 modern, introductory linear algebra
textbooks to investigate presentations of matrix multiplication. Using
Harels (1987) textbook analysis framework, we examined the sequencing of
matrix multiplication and its accompanying rationale. We found two principal
sequences: one which first defines the operation as a linear combination of
column vectors before introducing the dot product method (LC to DP), and
another which invokes the dot product method before linear combinations (DP
to LC). The rationale for these two trajectories varied in interesting ways.
LC to DP demonstrates that solving a system of linear equations is equivalent
to solving its corresponding matrix equation Ax=b. The rationale for DP to LC
was less focused, opting in several cases to postpone the explanation until
linear transformations are covered. We hope to initiate a discussion about
the effectiveness of and pedagogical implications for these two contrasting
approaches. 
Grand Mesa A 
Differentiated Student Thinking
While Solving A Distance Vs. Time Graph Problem Eric Pandiscio This study probes the thinking of students at different stages: a)
secondary students taking calculus, b) college students taking calculus, and
c) college students pursuing teacher certification taking a mathematics
course other than calculus. The study asks: 1) what is the nature of student
thinking when solving a graph problem, and 2) do students with different
levels of mathematical experience solve a graph problem differently? A pilot
investigation reveals many students estimate answers, even if they had
studied calculus. For the current study, data will be collected during Fall,
2013. Oral interviews will be conducted with a subset of the participants and
coded via Grounded Theory (Strauss & Corbin, 1990; Dick, 2005). This work
follows physics education (McDermott, Rosenquist & van Zee, 1987;
Thornton & Sokoloff, 1990; Kim & Kim, 2005), and mathematics
education (Chiu, Kessel, Moschkovich & MunchNunez, 2001; Moschkovich,
1996) that describe difficulties students have with graph interpretation. 
Grand Mesa B 
Undergraduate Students' Use Of
Intuitive, Informal, and Formal Reasoning To Decide On The Truth Value Of A
Mathematical Statement Kelly Bubp Although deciding on the truth value of mathematical statements is an
important part of the proving process, students are rarely engaged in making
such decisions. Thus, little is known about the ways in which students use
intuitive, informal, and formal reasoning to evaluate conjectures. In this
study, taskbased interviews will be conducted with undergraduate students in
which they will be asked to determine the truth value of five mathematical
statements on functions and relations. Students reasoning on these tasks
will be classified as intuitive, informal, or formal, and then further
categorized according to the findings of current research, with new
categories added as needed. This study should contribute to our understanding
of the ways in which students reason when dealing with uncertainty in the
proving process. Additionally, this study may suggest ways in which educators
can assist students in navigating the often difficult process of proving and
refuting mathematical statements. 
Grand Mesa C 
A Framework and a Study To
Characterize a Teachers Goals For
Student Learning Frank Marfai In this study, a secondary school teachers goals for student learning
were characterized using a framework that emerged from prior work. Observed
lessons spanning the use of both conceptually rich and conceptually poor
curricula were analyzed and lead to unexpected findings, suggesting that both
challenges and opportunities for professional development endeavors exist
that center around perturbing a teacher's goals. 
Chasm Creek B 
Instructors Beliefs On The Role Of
Calculus Kathleen Melhuish & Estrella Johnson In this report we will draw on the Characteristics of Successful
Programs in College Calculus data set in order to investigate instructor
beliefs about the role calculus plays. Specifically, in this preliminary report,
we have analyzed instructor interview transcripts in order to address the
question: How do instructors perceive the role of calculus at successful
fouryear universities? Our preliminary analysis has uncovered six emerging
themes. Each will be presented and illustrated with an instructors quote. 
Wind Star 
Mathematical Perceptions and
Problem Solving Of First Year Developmental Mathematics Students In A
FourYear Institution Anne Cawley I report initial findings of a
study that seeks to investigate the change in developmental (remedial)
mathematics students mathematical problem solving skills. I report on the
analysis of oneonone interviews with six students before a fourweek Intermediate
Algebra course. The ultimate goal is to see the extent to which their skills
changed after the course. Using a framework of reasoning developed by Lithner
(2000), I describe events in which one particular student shows plausible
reasoning and also reasoning based on established experience. I seek input
with regard to alternative frameworks or analysis of the data that may help
me interpret the findings. 
11:20
– 11:50 am 
Session 19 – Contributed Reports 
Grand Mesa B 
Teaching Methods and Student
Performance In Calculus I Barbara Trigalet, Lisa Mantini & R. Evan Davis Classroom teaching in multiple sections of Calculus I at a large
comprehensive research university was observed and coded using the Teaching
Dimensions Observation Protocol (TDOP). Multiple teaching styles were
identified ranging from low engagement to moderate engagement to high
engagement sometimes including student group work. Student performance on two
coursewide uniform exams and on the Calculus Concept Inventory was analyzed
for any correlations with teaching methods. Significant correlations were
found with high engagement teaching styles on both the first exam and the
final exam. However, no significant correlations were found on the Calculus
Concepts Inventory, indicating that students may not have exerted much effort
on this assessment. 
Chasm Creek A 
Exploring Students Ways Of
Thinking About Sampling Distributions Aaron Weinberg The concept of a sampling distribution plays a central role in the
process of making statistical inferences. However, students typically
struggle to understand and reason about sampling distributions. This study
seeks to characterize the ways undergraduate students think about sampling
distributions in scenarios involving repeated sampling and making statistical
inferences. Eight students in an introductory statistics class worked on
problems involving sampling distributions during a semistructured interview.
A framework was developed based on their responses to describe the ways they
discussed and coordinated various aspects of the population and sampling
distributions by focusing on the processes of sampling and repeated sampling;
these descriptions suggest that explicitly coordinating particular aspects of
these processes may correspond to the robustness of students conceptions of
sampling distributions. 
Grand Mesa A 
Supporting Students To Construct
Proofs: An Argument Assessment Tool Martha Byrne & Justin Boyle Engaging students in the construction of proofs often does not include
conversations about what does and does not count as proof, to the detriment
of the students. The critiques of studentgenerated arguments should be
communicated in a language common to instructor and student; such a language
can be developed via an assessment tool that is accessible to both parties.
This paper describes the development of an argument assessment tool that will
be useful for instructors and researchers both to assess students and
participants ability to construct proofs and to communicate those
assessments. The tool is introduced and two assessed student arguments are to
illustrate the tools application. Future work with the argument assessment
tool will include its use in a classroom as an instructional tool for
establishing a common language for instructor and students and providing the
foundation for discussions about proof production. 
Chasm Creek B 
Students' Use Of Parameters and
Variables To Reason About Multivariable Functions Eric Weber The purpose of this paper is to characterize students ways of
thinking about parameters and variables to reason about the behavior of
multivariable functions. I focus on two single variable calculus students,
Lisa and Carl, as they participated in a sequence of semistructured
exploratory teaching interviews intended to gain insight into 1) their
approaches to reasoning about the behavior of single variable functions, and
2) what role those approaches played in their initial thinking about the
behavior of functions of two, three and four variables. The interviews
suggest that the students ability to move flexibly between thinking about a
functions variables as parameters allowed them to generalize their reasoning
patterns about functions of n variables and extend that to functions of n+1
variables. I argue that their ability to parameterize functions allowed them
to reason about functions for which they could not initially visualize
representations. 
11:50am–1:50
pm Grand Mesa DEF 
Lunch 
1:50 –
2:20 pm 
Session 20 – Preliminary Reports

Chasm Creek A 
Slope and Derivative: Calculus
Students Understanding Of Rates Of Change Jen Tyne Studies have shown that students have difficulty with the concepts of
slope and derivative, especially in the case of reallife contexts. I used a
written survey to collect data from 75 differential calculus students.
Students answered questions about linear and nonlinear relationships and
interpretations of slope and derivative. My analysis focused on students
understanding of slope as a constant rate of change and derivative as an
instantaneous rate of change, and what these meant in the context of the
problems. Preliminary results indicate that students have more success with slope
questions than derivative questions (McNemars test, p<0.03), and that
while students correctly use the slope of a linear relationship to make
predictions, they do not demonstrate an understanding of the derivative as an
instantaneous rate of change and an estimate of the marginal change. Plans
for a modified survey and interviews are in place for fall 2013. 
Chasm Creek B 
Student Understanding Of Linear
Independence Of Functions David Plaxco, Megan Wawro & Lizette Zietsman In this study, we present preliminary findings regarding student
understanding of linear independence of vectorvalued functions. Students
were given a series of homework questionnaires and participated in individual
and paired interviews. The researchers used grounded theory to categorize
student approaches for determining linear (in)dependence of functions. In
order to gain insight into students intuitive notions, data were collected
before any formal instruction about the definition of linear independence of
functions. The researchers describe initial analyses of student approaches,
conjecturing their treatment of vectorvalued functions at specific tvalues
or for varying t as a potentially beneficial lens of analysis. Students who
evaluated specific tvalues determined the linear independence of a set of
vectors in R2 rather than the linear independence of the set of functions,
themselves elements of a function space. The analytical construct of
process/object pairs (Sfard, 1991) could be a useful lens to explore this
distinction. 
Grand Mesa B 
Proof Conceptions Of College
Calculus Students Jon Janelle This study investigated 52 college Calculus students views about the
nature and purpose of mathematical proof and which forms of empirical
argument they perceived as valid and convincing proofs. Past studies of
student proof conceptions have primarily focused on three groups: students in
secondary geometry courses, preservice and inservice teachers, and advanced
undergraduate and graduate students who have received formal instruction in
the creation of deductive proofs. This study fills a gap in the literature by
examining students conceptions after the completion of a high school
geometry course, but before enrollment in a course focused on the creation
and evaluation of mathematical proofs. Survey questions and coding systems
were adapted from previous studies. Preliminary findings suggest that a
majority of college Calculus believe that the inspection of a few cases and
the testing of an extreme case are valid methods for proving mathematical
conjectures. 
Wind Star 
An Investigation Into Students Use
Of Given Hypotheses When Proving Kathleen Melhuish The mathematical practice of strengthening or weakening a theorem
requires careful attention to hypothesis and conclusion. Selden and Selden
(1987) reported that students often unintentionally weaken theorems raising
concerns of undergraduates attention to hypothesis. In this paper, I
consider both the prevalence of this error and what the practice of
strengthening/weakening a theorem may look like. A survey of prove/disprove
prompts was piloted with five graduate students. A subset of these prompts
was then given to undergraduates in an introductory group theory course.
Preliminary results indicate that the error of weakening the theorem is
prevalent amongst both populations. The graduate students participated in
followup interviews where they were prompted to strengthen/weaken
conjectures to further examine their attention to the hypotheses. In this
preliminary report, I will present the survey results and one graduate case
to illustrate what the practice of strengthening/weakening a theorem may look
like. 
Grand Mesa A 
The Effect Of 5 Minute Preview
Video Lectures Using Smart Board, Camtasia Studio, and Podcasting On
Mathematical Achievement and Mathematics SelfEfficacy Minsu Kim The purpose of this study is to examine the effectiveness of 5 minute
preview video lectures for each lecture using podcasting in terms of
mathematical achievement and mathematics selfefficacy in intermediate
algebra and college algebra courses at a university. Data from 128 students
in six sections collected for two semesters through first and final exams,
questionnaires, classroom observation checklist, and the Mathematics
SelfEfficacy Scale. The preliminary findings indicate no significant
difference on the mathematical achievement and mathematics selfefficacy
between the control group who did not watch the preview lectures and the
treatment group who watched the preview lectures while the treatment group
slightly developed their mathematics selfefficacy and abilities for mobile
technology. In addition, the treatment group was significantly satisfied with
the preview lectures. When the treatment group was divided into intermediate
low and high subgroups based on the first exam, the intermediate low subgroup
significantly improved their mathematical achievement. 
Grand Mesa C 
Characteristics Of Successful
Programs In College Calculus: Instructors Perceptions Of The Usefulness and
Role Of Instructional Technology Erin Glover &
Sean Larsen The CSPCC (Characteristics of
Successful Programs in College Calculus) project is a large empirical study,
investigating mainstream Calculus 1, that aims to identify the factors that
contribute to successful programs. The CSPCC project consists of two phases.
Phase 1 entailed largescale surveys of a stratified random sample of college
Calculus 1 classes across the United States. Phase 2 involves explanatory
case study research into programs that were identified as successful based in
part on the results of the Phase 1 survey. This second phase will lead to the
development of a theoretical framework for understanding how to build a
successful program in calculus and in illustrative case studies for
widespread dissemination. Technology was one of the topics we explored with
students, instructors, administrators, and other individuals that we
interviewed during our case study site visits. In this preliminary report, we
will focus on calculus instructors views on instructional technology. 
2:30 –
3:00 pm 
Session 21 – Contributed Reports 
Grand Mesa A 
Reinventing Permutations and
Combinations Elise Lockwood, Craig Swinyard & John Caughman Counting problems provide an accessible context for rich mathematical
thinking, yet they can be surprisingly difficult for students. While some
researchers have addressed these difficulties, more work is needed to uncover
ways to help students count effectively. In an effort to foster conceptual
understanding that is grounded in students thinking, we had two
undergraduate students engage in guided reinvention in a tensession teaching
experiment. In this experiment, the students successfully reinvented four
basic counting formulas. In followup problems, combinations proved to be the
most problematic for them, however, suggesting that the learning of
combinations may require special attention. In this presentation, we describe
the students successful reinvention, and we discuss potential reasons for
the students issues with combinations. We additionally present potential
implications and directions for further research. 
Chasm Creek B 
Calculus Students' Understanding Of
Units Allison Dorko & Natasha Speer Units of measure are critical in many scientific fields. Instructors
often note that students struggle with units, yet little research has been
conducted about the nature of these difficulties or why they exist. Area and
volume play important roles in calculus topics such as optimization, volumes
of revolution, and related rates, yet we do not know what understandings of
area and volume students bring with them to their study of these topics. We
used written surveys and interview data to investigate calculus students use
of units in area and volume computational problems. Only 26.6% of students
gave correct units for all tasks. These students understood arrays, dimensionality,
and rules of exponents. In contrast, students who struggled with units did
not. Common errors included misappropriation of length units and difficulty
identifying the units of computations that involved . Our findings are
similar to findings about elementary school students difficulties with
units. 
Chasm Creek A 
Proof Scripts As a Lens For
Exploring Proof Comprehension Rina Zazkis & Dov Zazkis We examine perspective secondary teachers conceptions of what
constitutes comprehension of a given proof and their ideas of how students
comprehension can be evaluated. These are explored using a relatively novel
approach, scripted dialogues. The analysis utilizes and refines MejiaRamos,
Fuller, Weber, Rhoads and Samkoffs (2012) proof comprehension framework. We
suggest that this refinement is applicable to other studies on proof
comprehension. 
Grand Mesa B 
A Typology Of Validating Activity
In Mathematical Modeling Jennifer Czocher Mathematical modeling tasks are
used to help students learn mathematics and also to improve their modeling
skills. Validating has been identified as the process by which students check
and revise their models, but little is known about when or how students
choose to do so. This study examined engineering students validating
activity and identified a typology of different kinds of validating activity
satisfying different roles in ensuring accuracy of the model. 
3:00 –
3:30 pm Atrium 
Coffee Break

3:30 –
4:00 pm 
Session 22 – Contributed Reports

Grand Mesa A 
On The Sensitivity Of Problem
Phrasing  Exploring The Reliance Of Student Responses On Particular
Representations Of Infinite Series Danielle Champney This study will demonstrate the
ways in which students ideas about convergence of infinite series are deeply
connected to the particular representation of the mathematical content, in
ways that are often conflicting and selfcontradictory. Specifically, this
study explores the different limiting processes that students attend to when
presented with five different phrasings of a particular mathematical task 
(1/2)^n  and the ways in which each phrasing of the task brings to light
different ideas that were not evident or salient in the other phrasings of
the same task. This research suggests that when attempting to gain a more
robust understanding of the ways that students extend the ideas of calculus
– in this case, limit – one must take care to attend to not only
students reasoning and explanation, but also the implications of the
representations chosen to probe students conceptions, as these
representations may mask or alter student responses. 
Grand Mesa B 
How Does Undergraduates'
Understanding of the Function Concept Evolve During The Course of a Semester? Eyob Demeke,
Vincent Mateescu & Anek Janjaroon Functions are a crucial topic
in the study of mathematics. Research has found that a lack of deep
understanding of functions is one of the main reasons why students struggle
in calculus (Eisenberg, 1991; FerriniMundy & Graham, 1991; Lauten,
Graham, FerriniMundy, 1994; McDonald, Mathews, & Strobel, 2000; Monk,
1994). In light of these studies, we investigate – using traditional
paperandpencil assessments, concept maps, and an interview – what
precalculus students understanding of functions is, to what extent students
have a repertoire of functions at their disposal, how students understanding
evolves over a semester, and what nontraditional assessments can tell us
about this understanding. We found that (1) As Williams (1998) suggested,
concept map assessments do reveal something that traditional assessments do
not; (2) participants have trouble giving nonexamples of functions, and (3)
there does not seem to be a major change in participants understanding of
functions over time. 
Chasm Creek B 
Naive Brouwerian Visions: A Study
Of Students Interpretations Of NonConstructive Existence Proofs Stacy A. Brown This paper shares findings from
a threephase study exploring students conceptions of non constructive
existence proofs. Data are used to illustrate students tendency to apply a
nave Brouwerian lens to nonconstructive proofs; that is, a perspective in
which learners proof conceptions are governed by a potentially subconscious
anticipation of construction, which enables the learner to construe proofs of
existence (be they constructive or nonconstructive) as providing actual
instances of (or algorithms for producing) mathematical phenomena. Questions
concerning researchers proof scheme inferences are raised. 
Chasm Creek A 
Three Conceptualizations Of The Definite
Integral In Mathematics and Physics Contexts Steven Jones Student understanding of the
integral is a topic of recent interest in undergraduate education. We are
just beginning to learn how different interpretations of the definite
integral influence student thinking in both mathematics and science
classrooms. This paper examines the relative productivity of three
conceptualizations of the definite integral in mathematics and physics tasks.
It appeared that the notion of the integral as an addition over many pieces
was especially useful for understanding applied problems. 
4:10 –
4:40 pm 
Session 23 – Contributed Reports 
Chasm Creek B 
The Ability To Reject Invalid
Logical Inferences Predicts Proof Comprehension and Mathematics Performance Lara Alcock, Toby
Bailey, Matthew Inglis & Pamela Docherty In this paper we report a study
designed to investigate the impact of logical reasoning ability on proof
comprehension. Undergraduates beginning their study of proofbased
mathematics were asked to complete a conditional reasoning task that involved
deciding whether a stated conclusion follows necessarily from a statement of
the form if p then q; they were then asked to read a previously unseen
proof and to complete an associated comprehension test. To investigate the
broader impact of their conditional reasoning skills, we also constructed a
composite measure of the participants performance in their mathematics
courses. Analyses revealed that the ability to reject invalid
denialoftheantecedent and affirmationoftheconsequent inferences
predicted both proof comprehension and course performance, but the ability to
endorse valid modus tollens inferences did not. This result adds to a growing
body of research indicating that success in advanced mathematics does not
require a normatively correct material interpretation of conditional
statements. 
Grand Mesa C 
Factors Associated With The Success
Of Female Mathematics Doctoral Students Emily Miller Although the gender gap in
participation in undergraduate mathematics has narrowed, disparities still
exist at the doctoral level. Only 30 percent of recent doctoral recipients in
mathematics were women (Hill, Corbett, & St. Rose, 2010). To increase
retention of women in mathematics doctoral programs, it is critical to study the
factors that are associated with their success. A survey was distributed to
142 female mathematics professors asking them to assess the impact of factors
that could have contributed to their success. Results point to changeable
factors that can be implemented to narrow the gender gap. Salient factors
include persistence and dedication, strong undergraduate preparation and
quality doctoral courses, and support from the doctoral advisor. Results show
that gender still has an impact on the experiences of the participants, but
there may be reason for optimism. Respondents who received their doctorates
more recently reported less gender discrimination. 
Grand Mesa B 
Deploying Problems Assessing
Mathematical Knowledge For Teaching As Tasks For Professional Preparation Yvonne Lai & Heather Howell Mathematical knowledge for
teaching (MKT) has been shown to be a measurable construct impacting
instructional quality and student outcomes. The primary examples that
educators have for MKT tasks are those that were designed and validated for
assessment purposes. It is not known to what extent features of a task that
support its use as assessment may support or hinder its use in instruction.
We examine this tension by studying the use of two such MKT tasks in a course
for prospective teachers. Key considerations for using MKT tasks in
professional preparation tasks were how the MKT task represents teaching
practice and the possible purposes of using that representation in teacher
education. 
Chasm Creek A 
Exploring Differences In Teaching
Practice When Two Mathematics Instructors Enact The Same Lesson Joseph Wagner &
Karen Keene Investigating teacher practice
at all educational levels has become an important research arena. We analyze
teacher practice by comparing two implementations of the same fragment of a
studentcentered curriculum by two mathematics professors. We highlight
differences in their practices and the consequent classroom results by
analyzing their participation in class discussions, and we show how
Schoenfelds (2011) resources, goals, and orientations framework may be used
to explain these differences. Using classroom and interview data, we identify
resources that each instructor believed he lacked, we highlight prominent
mathematical and social goals that each instructor held, and we infer
orientations toward teaching and learning mathematics that guided each
instructors practices. All of these in combination suggest explanations for
the observed differences in the implementations and class outcomes. We
believe that this analysis provides an important technique to understand and
improve teaching and learning at the undergraduate level in mathematics. 
4:45 –
5:15 pm Atrium 
Break (Cash Bar) 
5:15 –
8:30 pm Grand Mesa DEF 
Awards Banquet & Plenary SessionSpeaker:
Ron Tzur Promoting
Teachers and Students Learning To Reason Multiplicatively: A
UnitsandOperations Developmental Approach Abstract: This paper presents an
approach to the learning and teaching of multiplicative reasoning that
focuses on units and operations students may construct, and use, when solving
and posing mathematical problems. To explain learning, this approach
includes a 6scheme developmental framework rooted in studies on
childrens construction of multiplicative and divisional schemes. This
framework (a) distinguishes between two types of units—singletons (1s)
and composite—each possibly comprised of concrete, figural, or abstract
items, and (b) articulates advances in students ways of coordinating
operations on either or both unit types. To promote students learning and
teacher development, this approach foregrounds a student adaptive
mathematical pedagogy (STAMP; shorthand – adaptive teaching).
Adaptive teaching stresses the need to tailor goals for student learning
(what should we teach next?) and activities for accomplishing these goals
(how should we teach this?) to students available ways of operating on
various units. Specifically, teachers learn to design and implement
tasks for reactivating, and transforming, schemes that are both available to
the students and instigate conceptual pathways to the intended mathematics.
Data collected and analyzed in studies that employed this approach will
be presented to elucidate and substantiate how it can contribute to teacher
change and to student learning and outcomes (both those with learning
disabilities and their normalachieving peers). Implications of this approach
to teaching undergraduate students entering university mathematics courses as
well as prospective elementary teachers will be discussed. 