Abstracts

for the Tenth Special Interest Group of the Mathematical Association of America

on Research in Undergraduate Mathematics Education

Conference on Research in

Undergraduate Mathematics Education

DoubleTree Hotel, San Diego - Mission Valley, California

February 22 – February 25, 2007

This report will be about a DVD designed to generate discussion between
mathematicians about their students’ thinking. It shows students
from transition course working on proof problems. Half of the
screen shows the student and the other half shows subtitles and their
writing. Segments of video are interleaved with screens of
questions designed to prompt reflection on issues in mathematics
education such as students’: lack of appreciation for the role of
definitions in formal mathematics (e.g. Vinner, 1991); restricted
knowledge of examples (e.g. Moore, 1994); lack of proving strategies
(e.g. Weber, 2001). The availability of this DVD will be timely in the
UK context in particular, since there is an increased emphasis on
teaching qualifications for new lecturers but a lack of content-based
materials for subjects such as mathematics. In the presentation
we will show segments of the video and present early feedback from
mathematicians at a number of universities.

Advancing Mathematical Activity is a new construct that allows
mathematics educators to study students’ participation in the K-16
mathematics classroom. Rasmussen, Zandieh, King, and Teppo (2005)
examined students’ participation in mathematical activity as it
advances in sophistication as an alternative to Tall’s notion of
advanced mathematical activity (1994). By using the concept of
advancing mathematical activity, I interpret students’ mathematical
activity in one inquiry oriented differential equations class and use
the interpretation to evaluate their growing participation in the
mathematical community of practice (Wenger, 1998). The construct of
advancing mathematical activity identifies five types of activity that
students may participate in: symbolizing, algorithmatizing, defining,
justifying, and experimenting. The analysis of the participation
in the classroom yielded students enacting all but the defining
activity. This kind of analysis provides a better understanding of the
mathematical activity of student mathematicians developing at the
university level.

Recently there has been a growing focus on understanding Graduate
Mathematics Teaching Assistants (GMTAs), their teaching development,
and preparation; however, we know little about our GMTA population and
how mathematics departments are using them. Addressing this problem, we
conducted a nationwide, electronic, survey study of U.S. mathematics
departments, to determine the extent of GMTAs’ instructional
involvement. Findings reveal important di_erences between GMTAs’
responsibilities and GTAs’ in other disciplines. Further, they show
diversity among mathematics departments’ use of GMTAs, particularly in
three areas: GMTA Population Size, Cultural Diversity, and Autonomy.
Departmental variations along these variables can create different
social contexts, plausibly impacting GMTAs’ professional development
and social networking opportunities. This report will detail and
support these findings, answering the question, “What is the extent and
nature of GMTA involvement in college mathematics instruction?”

Preparing graduate students to teach can directly impact the quality of
college mathematics instruction. Many programs have been developed to
help Graduate Mathematics Teaching Assistants (GMTAs) be successful,
but almost no research underlies most of these programs. Little has
been done to understand these programs’ impact and structure. September
2006, we initiated a longitudinal research study, aimed at developing
and understanding one form of professional development: the use of
Video Observations with Peer-feedback Sessions (VOPS). Utilizing
qualitative methods, we seek to understand how VOPS can be structured
to foster social teaching networks and what variables impact its
implementation. Preliminary data highlight factors impacting session
discourse (e.g. teaching assignments and prior teaching experience);
they further suggest that VOPS works in small group settings and can
effectively engage GMTAs with prior teaching experience or diverse
teaching assignments. In this report, I will detail these findings and
discuss preliminary implications for such a program’s implementation.

This workshop will provide techniques for maximizing an individual's use of the new Math Gateway and the Math Digital Library provided by the MAA for all members. MAA's Math Gateway is a remarkable search portal designed to implement searching undergraduate mathematics materials within the National Science Digital Library (NSDL). The Math Gateway brings together collections with significant mathematical content and services of particular importance to the delivery and use of mathematics on the Web. Math Gateway also provides a home page for the user that not only keeps track of favorite resources, but allows for a discussion board and shared resources with selected individuals. The workshop will show the particulars of using the Gateway and support will be provided from the makers of the Gateway.

The Role of the Teacher in the Evolution of Conjecturing as a Social Norm

This paper describes the actions taken by a teacher in a differential
equations course who set out to develop a classroom culture
characterized by inquiry. One of the primary means for achieving this
goal was to allow for and even request, and then to investigate
student-generated conjectures. The data shows that a culture of inquiry
was developed where students freely communicated and then explored
their own conjectures in the classroom. Moreover, as a social norm, the
nature of the students’ conjecturing activity evolved over the course
of the 16-week semester. The data makes it clear that the teacher
played a pivotal role in both the initial negotiation of the social
norm of conjecturing and its subsequent evolution. The results of this
research help to further clarify and underscore the important role
which the teacher plays in instigating reform in mathematics education.

The purpose of the research is to investigate the cognitive mechanisms
involved in the construction and understanding of proof. The central
research question is: what are the conceptual mechanisms that
make it possible for mathematicians, mathematics instructors and
students to construct a notion of proof, given their existing knowledge
and experience? The data collected will include oral language, written
symbols, gesturing, drawing, formal graphing, and other modalities
utilized when teaching, talking about, and creating proofs. The primary
data source will be videotapes of mathematicians, students and
instructors in both interview and naturalistic settings. The analytic
framework draws from cognitive linguistics, which utilizes language as
well as other modalities to infer the unconscious conceptual mechanisms
and source domains that are involved in the construction of new
knowledge. Preliminary results from a pilot study indicate differences
in language and gesturing when university instructors discuss different
kinds of proofs.

Related rates problems are a source of difficulty for many calculus
students. There has been little research on the role of the mental
model when solving these problems. Three mathematicians were observed
solving three related rates problems. From the examination of their
solutions, a framework for the solution process emerged. The framework
is based on five phases: draw a diagram, construct a functional
relationship, relate the rates, solve for the unknown rate, and check
the answer for reasonability. Each phase can be described by the
content knowledge the problem solver accesses, the mental model that is
developed, and the solution artifacts that are generated.

Interviews of students entering college trigonometry reveal high levels
of math anxiety; having to remember numerical tables and an
overwhelming amount of formulas unrelated to meaningful representations
are to be blamed for such apprehension. Furthermore students often
perceive trigonometry as disconnected from other mathematical topics
with no transition from previously acquired knowledge. Technology has
proven efficient in enhancing students’ learning by allowing them to
rediscover mathematical properties through visual manipulations. In
particular research on using Graphing Calculators, Excel spreadsheets
or The Geometer’s Sketchpad in middle school and high school
trigonometry courses show positive results for achieving higher
conceptual understanding. In this project the researchers propose to
make effective use of a unique lab setting in order to investigate the
effect of the interactive geometry software, The Geometer’s Sketchpad,
in a college trigonometry course. We will present activities designed
to support meaningful explorations of trigonometric concepts from a
constructivist perspective.

The purpose of this study is to characterize the types of instructional
strategies used by college mathematics professors in courses in which
perspective secondary (grades 7-12) teachers enroll. In this
study, online surveys are sent to the combined membership list of the
AMS and MAA in the United States as a way to gather the data. The
findings from this study emphasize the importance of documenting the
instructional strategies of college mathematics professors as a way to
determine the extent to which a variety of teaching approaches have
been adopted.

The study of calculus requires an ability to understand algebraic
variables as generalized numbers and as functionally-related
quantities. These more advanced uses of variables are indicative of
algebraic thinking as opposed to arithmetic thinking. This study
reports on entering Calculus I students’ responses to a selection of
test questions that required the use of variables in these advanced
ways. On average, students’ success rates on these questions were less
than 50%. An analysis of errors revealed students’ tendencies toward
arithmetic thinking when they attempted to answer questions that
required an ability to think of variables as changing quantities, a
characteristic of algebraic thinking. The results also show that
students who more successfully demonstrated the use of variables as
varying quantities were more likely to earn higher grades in Calculus I.

In this presentation, we share findings from research on knowledge
graduate students have about how calculus students think about limits.
At the K-12 level, researchers have demonstrated that improving teacher
knowledge of student thinking is a powerful model for professional
development, leading to changes in teachers’ practices and improvements
in student achievement. We hypothesize that this extends to the college
level–that improving graduate students’ knowledge of student thinking
can improve outcomes for college students. The initial step we take in
this larger project is to investigate what graduate students know about
their students’ thinking in calculus. We focus in particular on the
limit because of the relatively substantial body of research on student
thinking and misconceptions in this area. We will present some
preliminary findings and also seek audience members’ suggestions for
how the methods can be revised to strengthen findings in subsequent
studies.

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

Power differences in mathematical conversations with an authority such
as a teacher or tutor create unintended effects that may alter or
interfere with a student's reasoning during problem
solving. Using ideas from Sfard and Oehrtman, we analyze
transcripts of an authority interviewing second semester calculus
students as they attempt to solve a covariational reasoning
problem. We present an analytical framework focused on three
basic areas of student behavior that arise when interacting with a
mathematical authority’s scientific use of language: adaptations
to the scientific use of language, probes of the interviewer for
cues of approval regarding these adaptations, and attempts to
save face if adaptations appear unapproved.

Current college mathematics curricula for general education courses in
the U.S. lead to a collegiate mathematics education that falls short of
meeting national social justice and cultural competence needs. We
examine common views of curriculum, explore what it might mean for
college level mathematics curricula to be culturally responsive,
illustrate the development process for a Liberal Arts Mathematics
course, and exemplify what learning from such a curriculum might
include. In describing how a culturally responsive curriculum might
look, we take the meaning of “curriculum” as a dialogic process
associated with situated praxis (Grundy, 1987). We end by outlining
some of the research questions that arise around the theory of
culturally responsive curricula in college mathematics.

Based on Thompson’s (1994a, 1994b) and Carlson, Jacobs, Coe, Larsen,
& Hsu’s (2002) work on covarational reasoning we developed a new
covarational framework. The framework consists of three dimensions of
conceptual analysis, but this paper will focus on the dimension of
mental action. We present a lesson module where teachers are given the
task of determining the possible shape of a bottle when they are
provided with a graphical representation of the height of water in a
bottle as a function of the volume of water. We observed recurring
conceptual techniques that were not captured by any of the descriptions
of the mental actions in the original framework developed by Carlson
et. al (2002). In this paper, we propose new parameterized versions of
the previous mental actions and provide a description and examples for
one of these new categories in contrast to the previous mental actions
described by the original framework.

There is a widespread belief in the mathematics education community
that students should be encouraged to avoid basing their level of
conviction in mathematical arguments on the authority of the argument’s
source. In this presentation we report an experiment which investigated
the role that authority plays in the argument evaluation strategies of
undergraduate students and research active mathematicians. Our data
show that both groups were more persuaded by an argument if it came
from an authority figure. The implications of this finding are
discussed. It is argued that the role of authority in mathematical
argumentation – both in terms of actual behaviour and of normative
behaviour –requires deeper scrutiny.

Mathematicians hold a variety of different notions concerning the
purposes and necessary or sufficient conditions for proof. Thus it is
not surprising that mathematics majors in an advanced calculus course
found their professors from different courses each exhibited diverse
expectations of proofs produced by the students. The unintended
consequence of these varied expectations was that the students
developed a notion of an active audience for the proof. Hence a proof
was determined acceptable based on for whom it was written. The purpose
of this talk is to discuss the development of an active audience in
proof writing, its consequences and open the discussion as to which
notions of proof it would benefit students to develop.

Just as doing mathematics creates opportunities to learn mathematics,
“doing teaching” creates opportunities to learn to teach. Nowhere is
this more applicable than for graduate students who have little or no
teaching training prior to their first teaching assignments. We report
on our analysis of how the research literature on teachers’ on-the-job
learning can be applied to the context of graduate student professional
development. We combine this analysis with our synthesis of findings
about the role of teachers’ knowledge about student thinking in shaping
instructional practices and student learning opportunities. Our
findings take the form of a framework, grounded in research on teacher
learning, to guide the design of activities andprograms to equip
graduate students with the skills and dispositions to inquire into and
learn from their teaching experiences.

Linear algebra poses a number of significant challenges for students
that need to be better understood in order to improve instruction and
student understanding. At the time of this seminar we will have
just begun a teaching experiment intended to explore these
challenges. This preliminary report session will be a “working
session” in which we bring together participants to examine and discuss
the potential for specific modeling tasks to help make the difficult
transition to the formalism of linear algebra. We anticipate that
this session will also provide many opportunities for us to have
extended discussions with interested participants at other times during
the conference.

Issues of transfer typically arise in collegiate math classrooms when
students don't perform as well as expected on "application" problems.
This paper challenges one's assumptions about transfer by introducing
the actor-oriented transfer (AOT) perspective. Three principles arising
from AOT research will be presented and instructional implications
discussed.

The purpose of the ongoing research is to contribute to the development
of a conceptual analysis of the formal definition of limit. The
research is developmental in nature, consisting of a three-step
iterative cycle designed to accomplish two purposes: 1) to provide the
participants with optimal opportunity to come to reason coherently
about the formal definition of limit and, in so doing, 2) to produce
empirical evidence that will enable the identification of what we term
the ‘conceptual entailments’ of students’ reasoning about the formal
definition of limit.

The project is designed to contribute to existing research on
undergraduate students’ learning and understanding of quotient groups.
This report focuses on the first of a series of teaching experiments
aimed at developing an instructional approach that supports the
reinvention of the quotient group concept. The results of this first
iteration of the design-research cycle will be presented, with an
emphasis on sharing both preliminary discoveries and new questions that
have arisen.

This study explores views of infinity of first-year university students
enrolled in a mathematics foundation course, prior to and throughout
instruction on the mathematical theory involved. A series of
questionnaires that focus on geometrical representations of infinity
was administered over the course of several weeks. Along with
investigating students’ naïve conceptions of infinity, this
enquiry also examines changes of those views as beliefs, intuition, and
instruction are combined. The findings reveal that students’
conceptions about the nature of points, for instance, prevented them
from drawing any correlation between numbers and points on a number
line. Furthermore, a preliminary theoretical analysis using an
APOS framework asserts that participants conceive of infinity mainly as
a process, that is, as a potential to, say, create as many points as
desired on a line segment to account for their infinite number.

Preliminary report on an interview study of 14 mathematics faculty in
different higher education settings about the ways in which they use
mathematics textbooks for instruction. The instructors vary in terms of
their teaching experience, their research interests, the type of
courses they teach, and whether they themselves have written
mathematics textbooks. The study is geared towards understanding the
role that textbooks could play in assisting instructors in developing
their mathematics teaching expertise.

Activity theory serves as theoretical framework for studying the different activities that surround textbook use by instructors. Preliminary results suggest that instructors' textbook use varies depending on the level of course taught (upper or lower division courses) and by the course content (applied or not). It is less clear that instructors see the textbook as a source for assisting them in improving their teaching, even though they recognize its usefulness for lesson planning.

This article gives a construct of a year long course sequence of math
content for future elementary teachers (pre-service teachers, or PSTs)
at a research University. The sequence is conceptual based and
integrates assessment into it. Four issues motivate the courses: (1)
the lack of conceptual understanding by the PST, (2) the
misunderstanding of the objective of the sequence by the PST, (3)
incorporating NCTM Principles and Standards as a central focus, and (4)
the need to give PSTs more confidence with mathematics and their
ability to teach themselves the mathematics of elementary school.
Throughout, we describe how we constructed the course and discuss the
importance of the different tools used in the course. With this change
in place, we discuss a measure of the effectiveness of this course on
PSTs, and conclude with our data analysis.

This presentation describes research undertaken by a partnership
including nine school districts, two institutes of higher education,
and a non-profit organization. The partnership has made major
revisions to course offerings and support systems for pre-service and
in-service mathematics teachers. After a review of the literature and
of existing undergraduate programs, and discussion between district and
IHE partners, a new track of the mathematics major designed
specifically for future middle school mathematics teachers was
developed at one of the institutes of higher education. The partnership
also offers seven mathematics content courses during the summer
available to both in-service teachers and pre-service teachers (these
may be taken for university credit or professional development
hours). Preliminary results in this presentation describe changes
in the mathematics content knowledge of the pre-service and in-service
teachers, the classroom practice of in-service teachers, and the
mathematics content knowledge of the middle school students.

This preliminary research report will present results of an exploratory
study on the role of an instructional unit on undergraduate students’
logical reasoning skills. The instructional unit is based on the
pragmatic reasoning schema and is designed to encourage the use of
mathematical reasoning on “permission and obligation” problems.
Namely, students will translate traffic and parking regulation signs
into logic statements and use these statements to analyze various
situations. Data will be collected from students enrolled in a
mathematics course for liberal arts majors and pre-service secondary
school teachers enrolled in a college geometry course. Pre- and
post-instructional tests will be administered and students’ responses
and ability to transfer skills between context-sensitive and syntactic
problems will be analyzed. The preliminary data will be used to
reassess the research methods and refine the instructional unit.
Classroom implications, as well as future di rections for the study,
will be discussed.

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

for the Tenth Special Interest Group of the Mathematical Association of America

on Research in Undergraduate Mathematics Education

Conference on Research in

Undergraduate Mathematics Education

DoubleTree Hotel, San Diego - Mission Valley, California

February 22 – February 25, 2007

Engaging Mathematics Professors in Discussions about
Learning via Annotated Video of Student Proof Attempts

Lara Alcock University of Essex lalcock@essex.ac.uk |
Keith Weber Rutgers University khweber@rci.rutgers.edu |
Anne Seery Rutgers University aseery85@eden.rutgers.edu |

Karen Allen Valparaiso University karen.allen@valpo.edu |

Mathematics
Teaching Assistants: Their preparation and current involvement in
university instruction

Jason K.
Belnap Brigham Young University belnap@mathed.byu.edu |
Kimberly
Allred Brigham Young University kim.allred@gmail.com |

Mathematics
Teaching Assistants: Using Video Observation with Peer-Feedback
Sessions for Professional Development

Jason Belnap Brigham Young University belnap@mathed.byu.edu |

Teaching
Transformations of Functions using Modern Dance: An Experiment Pairing
a Modern Dance Class with College Algebra

The researcher’s interest in finding ways to improve college algebra
has led to a study of learning styles of College Algebra students.
Since 77% of the surveyed college algebra students are more active
learners than reflective learners, a new approach is needed. A College
Algebra course was paired with Modern Dance to enable students to learn
both the dance and the math curricula, by using action to help
understand the mathematics. This presentation will focus on the ways of
teaching Transformations of Function in a kinesthetic way to the
students in the class. The presentation will discuss both the learning
that occurred and the student’s attitude toward the subject.. Ann D. Bingham Peace College abingham@peace.edu |

This workshop will provide techniques for maximizing an individual's use of the new Math Gateway and the Math Digital Library provided by the MAA for all members. MAA's Math Gateway is a remarkable search portal designed to implement searching undergraduate mathematics materials within the National Science Digital Library (NSDL). The Math Gateway brings together collections with significant mathematical content and services of particular importance to the delivery and use of mathematics on the Web. Math Gateway also provides a home page for the user that not only keeps track of favorite resources, but allows for a discussion board and shared resources with selected individuals. The workshop will show the particulars of using the Gateway and support will be provided from the makers of the Gateway.

Reaching
for Understanding with Example-Generation Tasks

This study is a contribution to the ongoing research in undergraduate
mathematics education. It focuses on example-generation tasks as a
methodology to probe students’ understanding of mathematics. It is
guided by the belief that better understanding of students’
difficulties leads to improved instructional methods. The study
introduces example-generation tasks as an effective data collection
tool to investigate students’ learning of mathematical concepts. In
particular, this study focuses on students’ understanding of the
concepts of linear algebra and pre-service and in-service teacher
understanding of the concepts of transcendental numbers.
Simultaneously, it enhances the teaching of mathematics by developing a
set of example-generation tasks that are a valuable addition to the
undergraduate mathematics education. Marianna
Bogomolny Southern Oregon University bogomolnm@sou.edu |
Tanya
Berezovski Simon Fraser University tberezov@sfu.ca |

Why Do
Teachers Need a Rich Understanding of Number: Lessons Learned from
Teachers’ Use of Standards-Based Whole Number Lessons

In this talk we will explore findings from the Whole Number Study.
These findings indicate that teachers may incorporate reform approaches
while maintaining particular beliefs or ideas about mathematics, even
when the pedagogical implications of these beliefs and ideas interfere
with student learning goals, as espoused in reform curricula. To
illustrate this finding, we will examine a particular mathematical idea
that 2nd grade teachers in our study enacted as a “mathematical rule.”
This rule, which we will refer to as the “right to left” rule
(RL-rule), manifests itself when students are asked or required to
operate on numbers from right to left (i.e., ones, tens, etc.). Having
examined teachers’ use and views on the RL-rule, we will then turn our
attention to the implications of this work for preservice teacher (PST)
mathematics content courses and the ways in which Lesson Reviews might
facilitate the exploration and examination of PSTs’ ideas and beliefs
about mathematics. Stacy A.
Brown University of Illinois at Chicago stbrown@uic.edu |
Alison
Castro University of Illinois at Chicago amcastro@uic.edu |

The Role of the Teacher in the Evolution of Conjecturing as a Social Norm

Mark Burtch The American University in Dubai mburtch@aud.edu |

The Effect
of Interactive Computer Laboratory Activities on a Large Liberal Arts
Mathematics Course

This talk will detail preliminary research conducted in the West
Virginia University (WVU) Liberal Arts Mathematics course.
Initial research centers on developing an assessment to be used as a
pretest and posttest, the focus of which is using mathematics to solve
everyday problems. Further research involves the effect of an
interactive computer laboratory component on student performance in and
satisfaction with the course. The control section will meet for
two lectures on core material and one on applications per week. A
second section will have two lectures on core material and one computer
laboratory meeting a week, in which students interactively explore
applications using technology. The two sections will be compared
using the pre/post assessment, a pre/post attitude survey, exams,
quizzes, Personal Response System questions, and attendance.
Matched pairs of students will also be compared using the measures
listed above, to control for possible differences between the two
sections.Frederick
Butler West Virginia University fbutler@math.wvu.edu |
Melanie
Butler West Virginia University mbutler@math.wvu.edu |

The
Important of Decentering in the Role of a Professional Learning
Community Facilitator

In this project we investigated the interactions among members of a
professional learning community (PLC) of secondary mathematics and
science teachers. We report on our investigations of the facilitator’s
role in promoting meaningful discourse among the learning community
participants. We describe meaningful discourse in this PLC context as
involving substantive conversations about aspects of knowing, learning
and teaching mathematics content. As our research has evolved we
recognized that facilitators who made efforts to understand the
thinking and perspective of the PLC members were better able to engage
the members of the community in meaningful conversations. We call this
form of engagement acts of decentering. The data revealed five
manifestations of decentering. We illustrate through vides how four
different PLC facilitators made shifts in their decentering as
revealed by their attention or lack of attention to a PLC member’s
perspectives or knowledge, and whether they decided to act on this
knowledge during communication. Marilyn
Carlson Arizona State University marilyn.carlson@asu.edu |
Stacey
Bowling Arizona State University stacey.bowling@asu.edu |
Larisa
Kalachykhina Arizona State University l_chaika@yahoo.com |

Kevin Moore Arizona State University stacey.bowling@asu.edu |
Kelli
Wopperer Arizona State University kelli.wopperer@asu.edu |

Extending
the Descriptive Powers of Heuristics and Biases

My research examines the descriptive powers of the framework of
subjective probability, introduced by Psychologists Amos Tversky and
Daniel Kahneman, on mathematical misconceptions that arise in areas
other than probability. Tversky and Kahneman referred to
subjective probability as a probability estimate of an event, either
given by a subject or inferred from her or his behavior, and described
specific heuristics and biases associated with probabilistic
inferences. In particular, this report examines undergraduate
prospective elementary school teachers' use of prime numbers when asked
to simplify a "large" fraction in a clinical interview setting.
Participants' approaches to the task are interpreted through the
framework of judgment under uncertainty: more specifically, the
heuristics of representativeness, availability and adjustment from the
anchor, as well as their respective biases. The results suggest that
participants' struggles associated with elementary number theory
originate in the use of subjective probability.Egan J. Chernoff Simon Fraser University egan_chernoff@sfu.ca |

Dissertation
Abstracts: Scientific Evidence Related to Teaching and Learning
Mathematics

The availability of internet library resources, including UMI’s archive
of digital dissertations (http://wwwlib.umi.com/dissertations), has the
potential to link studies and enhance what we are learning from
research on undergraduate mathematics education. It is now
possible to read full text dissertations that have been selected based
on keywords found in the dissertation’s abstract. A 6/30/06
search using the keyword “mathematics education”, for example, yields a
total of 7,598 dissertation abstracts; 342 of these represent degrees
awarded in 2004. Of these 342 dissertations, 115 were awarded by
the 46 institutions posted (6/29/06) on the SIGMAA-RUME web as
institutions offering doctoral programs in mathematics
education. If separated from the research conducted in k-12
settings, a partial view of the dissertation research focusing on
undergraduate mathematics (n=29) completed in one year emerges.
This preliminary report and discussion about dissertation research has
the potential to help understand and guide future research, practice,
and policy development.Karen B. Cicmanec Morgan State University kbcicmanec@earthlink.net |

Documenting
the Emergence of “Speaking in Meaning” as a Sociomathematical Norm in
Professional Learning Community Discourse

The purpose of this research is to describe the sociomathematical norm
of speaking in meaning as well as describe its emergence in a
professional learning community (PLC). Speaking in meaning is used to
illustrate the type of mathematical participation expected of the
teachers. Speaking in meaning implies that the teachers not only base
their responses and suggestions on mathematical concepts but also give
conceptual descriptions when giving their explanations. The data for
this study is currently being collected from a PLC whose members are
secondary mathematics and science teachers. Initial analysis
shows that they are beginning to develop criteria for what is
acceptable. For example diagrams used must be clearly labeled and any
operations performed must be explained conceptually (not just
procedurally). This research has the potential to contribute to the
theoretical construct of speaking in meaning as well as inform the
instructional design of PLC?s.Phil Clark Scottsdale Community College phil.clark@sccmail.maricopa.edu |
Marilyn
Carlson Arizona State University Marilyn.Carlson@asu.edu |
Kacie Koch Arizona State University koch@mathpost.la.asu.edu |

Angela Ortiz Arizona State University ortiz@mathpost.la.asu.edu |
Joshua
McDaniel Arizona State University Joshua.McDaniel@asu.edu |
Katerina
Panagiotou Arizona State University Katerina.Panayiotou@asu.edu |

Student Use
of Mathematical Reasoning in Quasi-Empirical Investigations Using
Dynamic Geometry Software

This study focuses on how students incorporate mathematical reasoning
into investigations using Dynamic Geometry Software (DGS), with a
special emphasis on how mathematical definitions are incorporated into
these investigations. Six upper-level undergraduate students from
a comprehensive midwestern university in the United States of America
were asked to use DGS to justify three geometrical assertions in
individual, semi-structured interviews. The students generally
incorporated correct definitions into their DGS investigations but had
difficulty parsing the mathematical statements and exhibited
difficulties similar to the ones experienced when using definitions in
proofs. Jeff Connor Ohio University connor@math.ohiou.edu |
Laura Moss Ohio University moss@math.ohiou.edu |

Mathematical
Knowledge for Teaching: The Case of Complex Numbers

The use of complex numbers occurs throughout mathematics, engineering,
and science and undergraduates learn to use complex numbers in a
variety of courses, including calculus, differential equations, and
more advanced courses in complex analysis. Yet our review of the
literature on the teaching and learning of complex numbers at all grade
levels has, to date, revealed no empirical studies focused on this
important mathematical terrain. The proposed presentation will report
on one of the first empirical studies on student learning of complex
number, conducted during the last three weeks of the Fall 2006 semester
in a capstone mathematics course for prospective secondary school
mathematics teachers at a large southwestern university.Elizabeth
Connor San Diego State University conner@rohan.sdsu.edu |
Chris
Rasmussen San Diego State University chrisraz@sciences.sdsu.edu |

Michelle Zandieh Arizona State University zandieh@asu.edu |
Michael Smith San Diego State University msmith25@gmail.com |

High
School Teachers’ Orientation to Problem Solving and Learning: Striving
for an Answer or for Understanding?

When faced with a problem, there are many ways to orientate oneself to
it. Are you looking for a specific answer or do you want to understand
the conceptual underpinnings of the problem? We will present findings
from a study of the science, technology, engineering, and mathematics
(STEM) process and dispositional behaviors of high school math and
science teachers while involved in a professional development program
consisting of four graduate courses aimed at changing teaching practice
to more inquiry-based methods. We will argue that one’s orientation to
problem solving has a significant impact on learning outcomes and, in
the case of teachers, teaching practice.Cynthia
D’Angelo Arizona State University cynthia.dangelo@asu.edu |
Michael
Oehrtman Arizona State University oehrtman@math.asu.edu |

Assessments
that Improve Proof Writing Skills

In this qualitative study, we investigate what and how assessments help
students improve their proof writing skills. To investigate these
questions, students who completed 2nd semester of an undergraduate
abstract algebra course were interviewed and also completed an
open-ended questionnaire. Data from the interviews were
transcribed and coded using grounded theory methodologies as described
by Strauss and Corbin (1998). The three themes that emerged as
contributors to the improvement of students’ proof writing skills are
practicing writing proofs, observing proofs being done by others, and
receiving feedback on proofs. The assessments with which these
themes were reported were identified. Students reported that
in-class proof presentations provided an opportunity to engage with all
three themes, and homework provided opportunities to practice writing
proofs and receive feedback on proofs. These results indicate
that assessments that involve these three themes should be used in a
classroom where improving students’ proof writing skills is an
objective. Casey Dalton University of Northern Colorado daltonw13@yahoo.com |
Nissa
Yestness University of Northern Colorado nissa.yestness@unco.edu |
Hortensia
Soto-Johnson University of Northern Colorado hortensia.soto@unco.edu |

Developing
and Applying a Taxonomy of Mathematical Knowledge-Expertise

A taxonomy for mathematical knowledge-expertise was developed during a
year-long study of students' understanding of proof across the
undergraduate mathematics major at a medium-size comprehensive
university. The taxonomy takes the form of a matrix with elements
adapted from science assessment and expertise theory. The matrix was
developed in order to more accurately describe the performance of 12
students and one faculty expert on a "proof-aloud" task. The
taxonomy matrix characterizes three stages of mathematical expertise
across six cognitive and two affective components of mathematical
knowledge. This paper will describe how the taxonomy was developed,
what it implies about teaching practice, and its subsequent application
to analyzing videotapes of mathematics majors from large research
university engaged in problem solving sessions.Jacqueline
M. Dewar Loyola Marymount University jdewar@lmu.edu |
Curtis D.
Bennett Loyola Marymount University cbennett@lmu.edu |

The
Importance of the Concept of Function for Developing Understanding of
First-Order Differential Equations in Multiple Representations

The research reported here investigates the question, what is the
nature of students’ understanding of first-order differential equations
[FODEs] in a modern course on ordinary differential equations [ODEs]?
Modern courses on ODEs emphasize analytical, numerical, and qualitative
solution methods and hence use graphical and algebraic representations
of ODEs and their solution functions. Sfard’s (1991, 1994) theory of
reification predicts that a deep understanding of function is a
necessary component upon which understanding of FODEs is built; in the
terminology of the theory one must have reified the concept of
function. The two case studies reported on in this talk show a contrast
in understanding that gives insight into the cognitive importance of a
reified notion of function to the development of FODE understanding in
a modern, multi-representational approach to their study. John E. Donovan II University of Maine john.donovan@maine.edu |

Laurie D. Edwards Saint Mary's College of California ledwards@stmarys-ca.edu |

Nicole Engelke California State University, Fullerton nengelke@fullerton.edu |

Axelle P.
Faughn California State University Bakersfield afaughn@csub.edu |
Terran
Felter Murphy California State University Bakersfield tfelter@csub.edu |

Instructional
Strategies Used By College Mathematics Professors in Courses in Which
Perspective (grades 7-12) Teachers Enroll

Kelly Finn University of Iowa kelly-f-finn@uiowa.edu |

Intellectual
Need in High School Classrooms

Intellectual need, a key part of the DNR theoretical framework, is
posited to be necessary for significant learning to occur. I will
introduce the concept of intellectual need and explore ways in which it
can be absent from mathematics classrooms. A case study of two
high school algebra teachers whose classes were videotaped illustrates
several categories of activity in which students feel little or no
intellectual need. After identifying actions by the teacher that
may contribute to this “problem-free activity,” I suggest alternative
treatments of some lessons and more general ways to stimulate
intellectual need.Evan Fuller
University of California, San Diego edfuller@ucsd.edu |
Jeffrey M.
Rabin University of California, San Diego jrabin@ucsd.edu |
Guershon
Harel University of California, San Diego gharel@ucsd.edu |

How
Conceptually Important Calculus Ideas and Connections Emerge from
Collaborations

As part of a three-semester teaching experiment in calculus, 22
university students collaboratively explored open response tasks. We
analyze videodata of students exploring the Quabbin Reservoir Task and
presenting their ideas. We study emergence of conceptually important
calculus ideas (CICIs) and connections amongst students’ experience,
other classes, previous tasks, and CICIs. Based upon our preliminary
research, we expect that horizontal and vertical mathematizing,
conceptual blending, metonymy, and linguistic invention will be helpful
in categorizing ways CICIs and explicit connections are brought forward
through students’ explorations. However, these ideas alone are not
sufficient to understand the genesis of CICIs or connections. Other
important ideas emerging from the analysis are that justification plays
a crucial role in the building of CICIs and connections, student
questions often focus inquiry and justification, and speculation about
what “they” (the professors) want or questioning how current
explorations fit with previous class experiences promote connections
and formalizing.Hope Gerson
Brigham Young University hope@mathed.byu.edu |
Janet
Walter Brigham Young University jwalter@mathed.byu.edu |

Susan S.
Gray University of New England Biddeford, Maine sgray@une.edu |
Barbara J.
Loud Regis College Weston, Massachusetts barbara.loud@regiscollege.edu |
Carole P.
Sokolowski Merrimack College No. Andover, Massachusetts carole.sokolowski@merrimack.edu |

Beste
Gucler Michigan State University guclerbe@msu.edu |
Natasha
Speer Michigan State University nmspeer@msu.edu |

Jon
Hasenbank University of Wisconsin - La Crosse hasenban.jon@uwlax.edu |
Ted Hodgson Montana State University hodgson@montana.edu |

David
Hasson San Francisco State University davidhasson@yahoo.com |
Eric Hsu San Francisco State University erichsu@math.sfsu.edu |

Shandy Hauk
University of Northern Colorado hauk@unco.edu |
Mark K.
Davis Learning Helix and Center for Learning and Teaching in the West m.k.davis@comcast.net |

Nate
Hisamura Arizona State University nhisamura@cox.net |
Arlene
Evangelista Arizona State University arlene@mathpost.asu.edu |
Michael
Oehrtman Arizona State University oehrtman@math.asu.edu |

Matthew
Inglis University of Warwick m.j.inglis@warwick.ac.uk |
Juan Pablo
Mejia-Ramos University of Warwick j.p.mejia@warwick.ac.uk |

New
Definition, Old Concepts: Exploring the Connections in Combinatorics

Definitions are one of the most important parts of mathematics. The
importance of understanding formal definitions in teaching and learning
mathematics has been discussed in the literature. This research
explores students? understanding of a new definition and the
connections they make from this new definition to the concepts they
have previously learned. Eight first year undergraduate students
participated in this study. Before the interview, the participants were
given a definition that they have never seen before. During the
interview, they were presented with a set of tasks, which examined
their understanding of the new definition as well as their general
understanding of elementary combinatorics. In this report, I will
examine the ways that these students attempt to understand this new
definition, and how they use this new definition to solve one of the
problems that was given to them during the interview. Shabnam Kavouian Langara College skavousi@langara.bc.ca |

Jessica Knapp Pima Community College knapp@mathpost.asu.edu or jlknapp@pima.edu |

Teaching
Assistants Learning to Teach: Recasting Early Teaching Experiences as
Rich Learning Opportunities

David Kung St. Mary's College of Maryland dtkung@smcm.edu |
Natasha
Speer Michigan State University nmspeer@msu.edu |

Record-of /
Tool-for Transitions: Significant Shifts in the Way Students Use
Notational Systems

In this paper, the notion of a record-of / tool-for transition is
introduced. This transition involves an important shift in the role
played by a form of notation. This transition occurs when a form of
notation that has previously been by students primarily as a way to
record their informal mathematical activity begins to be used as tool
by students to support more formal mathematical reasoning.
Examples from both an abstract algebra context and a differential
equation context will be used to illustrate the notion of a record-of /
tool-for transition. A theoretical discussion will situate the notion
of a record-of / tool-for transition relative to other constructs
within the theory of realistic mathematics education (RME). Sean Larsen Portland State University slarsen@pdx.edu |
Karen
Marrongelle Portland State University karenmar@pdx.edu |

Christine
Larson Indiana University larson.christy@gmail.com |
Jill
Nelipovich San Diego State University jnelipov@ucsd.edu |
Chris
Rasmussen San Diego State University chrisraz@sciences.sdsu.edu |

Michael
Smith San Diego State University msmith25@gmail.com |
Michelle
Zandieh Arizona State University zandieh@asu.edu |

The Transfer
of Learning: A Simple Shift in Assumptions with Multiple Implications
for Teaching Undergraduate Mathematics

Joanne Lobato San Diego State University lobato@saturn.sdsu.edu |

Research
on Students’ Reasoning about the Formal Definition of Limit: An
Evolving Conceptual Analysis

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

Elise
Lockwood Portland State University elockwoo@pdx.edu |
Sean Larsen Portland State University slarsen@pdx.edu |
Joanna
Bartlo Portland State University joannamd@aol.com |

Ami Mamolo Simon Fraser University amamolo@sfu.ca |

Vilma Mesa University of Michigan vmesa@umich.edu |

Activity theory serves as theoretical framework for studying the different activities that surround textbook use by instructors. Preliminary results suggest that instructors' textbook use varies depending on the level of course taught (upper or lower division courses) and by the course content (applied or not). It is less clear that instructors see the textbook as a source for assisting them in improving their teaching, even though they recognize its usefulness for lesson planning.

Richard
Millman University of Kentucky millman@ms.uky.edu |
Matthew
Wells University of Kentucky mwells@ms.uky.edu |

Bernadette Mullins Birmingham-Southern College bmullins@bsc.edu |

Mika Munakata Montclair State University munakatam@mail.montclair.edu |

Kinesthetic
Experiences in the Process of Making Sense of Formal Equations

In this paper we investigate how kinesthetic experiences can play the
role of “bridges” that experientially bring together partial results
obtained by symbol manipulation with certain “states of affairs” that
students have engaged with physically. The selected interview episodes
show that kinesthetic experience can transfer or generalize to the
building and interpretation of formal, highly symbolic mathematical
expressions. The paper analyzes selected episodes from open-ended
individual interviews with three students who had taken, or were
taking, a class on differential equations. In the interviews students
engaged in a number of different tasks involving a physical tool called
the water wheel. Ricardo
Nemirovsky San Diego State University nemirovsky@sciences.sdsu.edu |
Chris
Rasmussen San Diego State University chrisraz@sciences.sdsu.edu |

Framing an
Interdisciplinary Multi-Year Professional Development Project for
Secondary Mathematics and Science Teachers

This session will describe the design research on a long-term
interdisciplinary professional development program for secondary
science and mathematics teachers. The intervention involves intensive
work with university faculty and school leaders in mathematics,
physics, chemistry, biology, geology, and engineering to develop a four
course graduate course sequence. A central challenge in this work is to
maintain coherence in the implementation, evaluation, and research. We
aim to achieve this coherence through a focus on unifying reasoning
patterns, unifying process behaviors, and unifying dispositional
behaviors across the project disciplines. In this research report, we
provide an overview of the development of our theoretical frameworks in
two of these three areas, focusing on covariational reasoning as a
unifying reasoning pattern and establishing an inquiry orientation as a
unifying process behavior. We illustrate through use of video data how
our frameworks and interventions have been adapted over three
iterations of the intervention. Michael
Oehrtman Arizona State University oehrtman@math.asu.edu |
Marilyn
Carlson Arizona State University marilyn.carlson@asu.edu |

Investigating
Linear and Exponential Reasoning of Students in a Reformed College
Algebra Course

Researchers hoped to show that students in an inquiry-oriented,
application-based course dealing with college algebra topics could gain
in their conceptual underpinnings without sacrificing mechanical
skills. Data from the study confirm one hypothesis and refute the
other. Specifically, the study investigated the achievement of
two clusters of students on two measures of linear and exponential
reasoning. Students in a traditional College Algebra course
outperformed students in a “reformed” Algebraic Modeling course on a
standardized instrument emphasizing procedural and mechanical
skills. Students in the Modeling course had significantly higher
scores on the instrument developed for the study, that emphasized
conceptual understandings. One conclusion drawn by the study is
that a modified curriculum and pedagogy can produce alternate outcomes,
while leaving open the question of what types of outcomes are most
desirable.Eric A. Pandiscio University of Maine eric.pandiscio@umit.maine.edu |

Roles of
Revoicing in the Inquiry-Oriented Mathematics Class: The Case of
Undergraduate Differential Equations Class

Recently mathematics educational researchers have taken an increasing
concern in the teacher¹s discourse move, which is defined as a
deliberate action taken by a teacher to participate in or influence the
discourse in the mathematics classroom (Krussel, Edwards, &
Springer, 2004). This study explored the roles of revoicing in the
undergraduate inquiry-oriented mathematics class in the perspective of
teacher¹s discourse move. The data for this analysis came from
four classes about phase portrait of the system of differential
equations with initial value from a large state university. We
particularly analyzed revoicing linked with questioning, telling, and
directing through the result of coding of teacher¹s discourse
move. The results show that revoicing has the following roles: bonding
students¹ response to the teacher¹s discourse move -
questioning, telling, or directing; providing students the ownership of
knowledge; providing students the springboard for further thinking. Jae Hee Park Seoul National University udmji@snu.ac.kr |
Oh Nam Kwon Seoul National University onkwon@snu.ac.kr |
Mi Kyung Ju Silla University mkju@silla.ac.kr |

Chris
Rasmussen San Diego State University chrisraz@sciences.sdsu.edu |
Karen
Marrongelle Portland State University karenmar@pdx.edu |

Teacher’s
Questioning in Argumentation: The Case of an Inquiry-Oriented
Differential Equations Class

The research builds on the Inquiry-Oriented Differential Equations
(IO-DE) project to develop a model for how it is that teachers create
and sustain inquiry-oriented mathematics classrooms to support
students’ learning mathematics in powerful and deep ways. For the
purpose, we have conducted a semester long classroom teaching
experiment. The analysis of this research focuses on the teacher’s
questioning in the context of students’ argumentation in an IO-DE
course. The analysis identifies five types of questions used by the
teacher in the IO-DE: Evaluative questions, Requests to explain
thinking, Requests to justify thinking, Requests to check or assess
student progress, and Clarifying questions. This research describes the
roles of these questions in the context of students’ argumentation to
reveal how the teacher strategically applies questions for the
construction of mathematics. This research implies that teachers’
knowledge of questioning is of essence for the improvement of
mathematics instruction. Jung Sook
Park Seoul National University pjungsook@hanafos.com |
Kyoung Hee
Cho Seoul National University cho0114@snu.ac.kr |
Oh Nam Kwon Seoul National University onkwon@snu.ac.kr |

Mi Kyung Ju Silla University mkju@silla.ac.kr |
Chris
Rasmussen San Diego State University chrisraz@sciences.sdsu.edu |
Karen
Marrongelle Portland State University karenmar@pdx.edu |

Justification
and Proof Schemes in High School Algebra Classrooms

We discuss the types of justifications offered by two high-school
algebra teachers, based on classroom observations during a two-year
period. These are classified using the proof scheme taxonomy of Harel
and Sowder and illustrated with examples. Justifications based on
authority or on empirical evidence predominate over deductive ones. The
type and quality of justifications offered can be limited by teachers’
pedagogical knowledge as well as their mathematical (content) knowledge.Jeffrey M.
Rabin University of California, San Diego jrabin@ucsd.edu |
Evan Fuller
University of California, San Diego edfuller@ucsd.edu |
Guershon
Harel University of California, San Diego gharel@ucsd.edu |

Using
Videocases to Bridge the Gap Between Heuristic and Formal Proofs

This preliminary report describes how videocases can be used to help
students overcome difficulties in learning to produce mathematical
proofs. The study was conducted in the context of a bridge-type
proof course at the university level. During class, students
watched and discussed videos of “competent” problem solvers talking
aloud while proving several statements that are normally difficult for
students at this level. The goal of the session was to help
students connect informal and formal aspects of proof. During the
following class, students were interviewed in both small group settings
and a whole class setting about their reaction to watching the
video. This talk will center on the relative advantages and
disadvantages of this type of intervention, and will solicit ideas for
improvement for the third iteration.Manya
Raman Rutgers University mjraman@rci.rutgers.edu |
James
Sandefur Georgetown University sandefur@georgetown.edu |

Helping
Students Learn How to Make Sense of Mathematical Statements Involving
Multiple Quantifiers

This preliminary report focuses on an effort to understand how students
make sense of mathematical statements involving multiple quantifiers
and to help students learn how to do so in a way that is consistent
with mathematical convention. Results are presented from a collection
of mini teaching experiments conducted with the aim of developing an
instructional approach that builds on students’ own mathematical
activity.Sonya
Redmond Portland State University sonyaredmond@comcast.net |
Sean Larsen Portland State University slarsen@pdx.edu |

Access to
Algebra: Comparative Study of High School Math Students Using Distance
Learning at Readiness with College Algebra Classroom Students

This is a preliminary report on a study comparing student achievement
between traditional and distance learning versions of the same college
algebra course. The traditional cohort is a group of 37 college
freshmen from the Appalachian area while the distant cohort is 38
rural, primarily Appalachian high school seniors. We will discuss the
efforts to provide a comparable content and instructional experience to
both groups. All students are completing the same coursework.
Examinations are uniformly graded. The local tutoring support for
traditional students is matched by real-time e-tutoring for the distant
cohort. In addition to achievement, the study is designed to identify
critical elements of an effective model for a distance learning college
algebra course. We use an exploratory investigation with a concurrent
design, involving examining student scores and survey data. The mixed
methods used in this study allowed for additional insights into the
attitude of distance learning in.Lee Alan Hanawalt Roher University of Kentucky lroher@ms.uky.edu |

Intermediate
Mechanics Students’ Coordinate System Choice

As part of ongoing research into cognitive processes and student
thought, we investigate the interplay between mathematics and physics
resources in intermediate mechanics students. In the mechanics
course, the selection and application of coordinate systems is a
consistent thread. Students start the course with a strong
preference to use Cartesian coordinates. In small group interviews and
in homework help sessions, we ask students to define a coordinate
system and set up the equations of motion for a simple pendulum where
polar coordinates are more appropriate. Using a combination of
Process/Object Theory[1] and Resource Theory[2], we analyze the video
data from these encounters. We find that students sometimes
persist in using an inappropriate Cartesian system. Furthermore,
students often derive (rather than recall) the details of the polar
coordinate system, indicating that their knowledge is far from solid. Eleanor C.
Sayre University of Maine le@zaposa.com |
John E.
Donovan II University of Maine jdonovan@math.umaine.edu |

Calculus
Students’ Assimilation of the Riemann Integral

We will report on results from a teaching experiment that used the
ideas of approximation (finding over and under estimates, determining a
bound on the error, and finding an approximation accurate to within any
predetermined bound) to develop a strong conceptual understanding of
the structure of the Riemann integral with college calculus
students. Students were able to successfully assimilate the
Riemann integral structure into an already established limit structure
(via approximations). Their struggles were mainly concentrated in
areas where the particulars of Riemann sums departed from the limit
structures students had developed while learning about limits of
functions, limits of sequences, and the definition of the derivative.Vicki
Sealey Arizona State University vicki@mathpost.asu.edu |
Michael
Oehrtman Arizona State University oehrtman@math.asu.edu |

Mathematical
Sophistication among Preservice Elementary Teachers

This study explores the ways in which eleven preservice elementary
teachers used a web-based teacher resource to apply a mathematical
definition, to correct a procedural error in arithmetic, and to make
sense of a story requiring the multiplication of fractions. In our
analysis we propose a framework to compare the behaviors and values
expressed by our participants with the values and norms of the
mathematical community. This analysis suggests that many preservice
elementary teachers are profoundly mathematically unsophisticated. In
other words, they displayed a set of values and avenues for doing
mathematics so different from that of the mathematical community, and
so impoverished, that they found it difficult to create fundamental
mathematical understandings.Carol E.
Seaman University of Wisconsin Oshkosh seaman@uwosh.edu |
Jennifer
Earles Szydlik University of Wisconsin Oshkosh szydlik@uwosh.edu |

The practice
of Teaching Collegiate Mathematics: An Important but Missing Topic of
Research

In this presentation, we summarize our search for empirical research on
collegiate mathematics teaching practices. Where research about
teaching is relatively common, descriptive empirical research on what
collegiate mathematics teachers actually do as they teach is virtually
non-existent. Our claim is based on a review of peer-reviewed journals
where research on collegiate mathematics teaching is published. Because
such research is needed (to illustrate innovative practices, promote
more research on practice, and support beginning teachers? learning
about teaching) we also propose a framework for studying collegiate
teaching practice. Its development was informed by a similar frame for
K-12 teaching (NCTM, 1991) and our analysis of teachers’ actions and
decision-making in the most common instructional format, lecture
presentation. We assert that college mathematics teachers make many
important decisions as they plan and carry out their instruction and
much can be learned by examining those practices.John P.
Smith III Michigan State University jsmith@msu.edu |
Natasha
Speer Michigan State University nmspeer@msu.edu |
Aladar
Horvath Michigan State University horvat54@msu.edu |

The
Calculus Project: What Does it Mean to Understand the Calculus?

This research is part of a larger investigation designed to explore
what it means for students to understand the Calculus. The initial
phase of the study seeks to comprehensively examine the
perceptions of well-known experts in the fields of mathematics to
identify the concepts and skills important to the calculus curriculum
and delineate a number of mathematical problems these same experts
believe could be used to assess students' level of understanding of
those concepts and skills. Thirty participants will be identified
and interviewed to collect information related to their knowledge and
beliefs about student understanding, the calculus curriculum, and
assessment. Interview data will be transcribed and analyzed using
methods of categorical content analysis to extract themes and patterns.
Initial findings from the analysis of the interview data will be
reported. Discussions of the research questions, design, and
preliminary findings will help to refine the future direction of the
study.Kimberly S.
Sofronas Emmanuel College sofronki@emmanuel.edu |
Nick
Gorgievski Nichols College nick.gorgievski@nichols.edu |
Larissa B.
Schroeder University of Connecticut larissa.schroeder@huskymail.uconn.edu |

Chris
Hamelin University of Connecticut chris.hamelin@huskymail.uconn.edu |
Charles
Vinsonhaler University of Connecticut vinsonhaler@math.uconn.edu |
Thomas C.
DeFranco University of Connecticut tom.defranco@uconn.edu |

Interfering
Knowledge: How it Hinders Proof-Writing

Weber (2001) describes four types of strategic knowledge (Hart, 1994)
that assist students with proof-writing. They are knowledge of: (1)
proof techniques, (2) important theorems, (3) usefulness of theorems,
and (4) timing for syntactic strategies. In comparing doctoral and
undergraduate students’ abstract algebra proof constructions, Weber
discovers that such strategic knowledge not found in undergraduates is
possessed by graduate students. In this report we describe somewhat
conflicting evidence to Weber’s results. Our findings indicate that
although our undergraduate students enrolled in abstract algebra
possess proof techniques, know the important theorems, and are able to
see the usefulness of a theorem, they continue to struggle with
proof-writing. One contributing obstacle to this phenomena is symbol
manipulation, which Weber briefly discusses. We define the second
obstacle as interfering knowledge; this is knowledge possessed by a
student that conflicts with the new content or interferes with the
ability to carry out a proof.Hortensia
Soto-Johnson University of Northern Colorado hortensia.soto@unco.edu |
Nissa
Yestness University of Northern Colorado nissa.yestness@unco.edu |
Casey Dalton University of Northern Colorado daltonw13@yahoo.com |

From Test
Cases to Special Cases: Four Undergraduates Unpack a Formula for
Combinations

We present preliminary findings of a case study through which we
sought, through detailed analysis of four students’ arguments, to
distil a set of analytic constructs that might help make clearer sense,
in general, of how arguments conveyed through special cases might
support assertions that are understood to hold in general. We began
analysis from a particular standpoint: to focus fundamentally on the
learners’ representations and on how they reasoned from them. We found
it helpful to distinguish two perspectives to guide the subsequent
analysis. On the one hand, we direct detailed attention to how learners
reason, most especially on how they organize the logic of their
arguments. On the other hand we seek to understand the learners’
representations through the way they structured them, and through how
such structures might be reshaped or reframed over time.Bob Speiser Brigham Young University speiser@byu.edu |
Chuck Walter Brigham Young University walterc@mathed.byu.edu |

Eigenvalues
and Eigenvectors: Formal, Symbolic and Embodied Thinking

Many beginning university students struggle with the new approaches to
mathematics that they find in their courses due to a shift in
presentation of mathematical ideas, from a procedural approach to
concept definitions and deductive derivations, with ideas building upon
each other in quick succession. This paper highlights this situation by
considering some conceptual processes and difficulties students find in
learning about eigenvalues and eigenvectors. We use the theoretical
framework of Tall’s three worlds of mathematics, along with
perspectives from process-object and representational theory. The
results of the study describe the thinking about these concepts of
groups by first and second year university students, and in particular
the obstacles they faced, and the emerging links some were constructing
between parts of their concept images formed from the embodied,
symbolic and formal worlds. We also identify some fundamental problems
with student understanding of the definition of eigenvectors that lead
to problems using it, and some of the concepts underlying the
difficulties.Sepideh
Stewart The University of Auckland sepideh@math.auckland.ac.nz |
Michael O.
J. Thomas The University of Auckland m.thomas@math.auckland.ac.nz |

Assessing
Peer Interactions in Secondary Science and Mathematics Teacher
Professional Learning Communities

This paper discusses whether the level of inquiry orientation in
teachers’ Professional Learning Community (PLC) provides a means to
characterize the group’s effectiveness. The PLCs in this study consist
of high school mathematics and science teachers participating in a
professional development project. By focusing on the interactions
between the teachers, we hypothesized that the inquiry orientation (the
group’s inclination to engage in problematic issues, to address common
problematic issues, and to work to bring resolution to the problematic
issues) could be a good predictor of effectiveness. We use this
perspective to contrast two PLC groups: one that was considered to be
very ineffective and one that was considered to have productive
discourse. The inquiry orientation of the members of the PLC seems to
have a strong influence on the discourse within the group.Judy Sutor Arizona State University judy.sutor@asu.edu |
Manuel Garay Arizona State University manuel.garayvalenzuela@asu.edu |
Michael
Oehrtman Arizona State University oehrtman@math.asu.edu |

Proof
Learning: From Truth Towards Validity

Pushing more radically Duval's (1991) research orientations, and taking
into account reflexions initiated in Author (2005) regarding the
obstacle we identify as ‘hindering truth value’, we designed tasks in
which pupils fit into an oriented graph the given propositions of a
geometrical proof. An experiment involving a sequence of three such
tasks was conducted in the spring of 2005 and of 2006. Among the
conclusions drawn from the data that were gathered: the role played by
rules of inference is underestimated by pupils; teams that took the
rules into account at each step were better achievers than teams that
(tried to) set the rules afterwards; teams that worked from the end
forward to the beginning were also better and faster achievers; the
organizational work required by the proposed tasks contributed to
fostering pupils' understanding of the mechanisms that rule the
deductive structure.Denis Tanguay Université du Québec à Montréal tanguay.denis@uqam.ca |

Affecting
Secondary Mathematics Teachers’ Instructional Practices by Affecting
their Mathematical Knowledge

Seven papers, spread over two sessions, will report on the first 18
months of a five-year project that aims to improve secondary
mathematics teachers’ instruction while at the same time producing
theoretical understandings of the intervention’s effects on teachers
beliefs, values, and knowledge. One paper will give an overview of the
project and a broad summary of its results. Five other papers will
report on the effects of covariational approaches to functions on
teachers’ reasoning, on making meaning as a means for professional
development, on decentering as a critical component of teacher
practice, on teaching for meaning in trigonometry, and on an extension
of the project into one classroom.Pat
Thompson Arizona State University pat.thompson@asu.edu |
Marilyn
Carlson Arizona State University marilyn.carlson@asu.edu |
Irene Bloom Arizona State University irene.bloom@asu.edu |
Sharon Lima
Arizona State University lamamushka@gmail.com |

Chris Miller Arizona State University christina.miller@asu.edu |
Jeff Rodel Arizona State University jeffrey.rodel@asu.edu |
Carlos
Castillo-Garsow Arizona State University cwcg@asu.edu |
Ana E. Lage Arizona State University Ana.Lageramirez@asu.edu |

A
Framework for Evaluation Online Calculus Courses

As computers become a part of everyday life, many students are
receiving instruction in online environments. Universities,
corporations, and even hobbyists are producing online course materials
made available for general public use. What differentiates these
courses, and what features of an online instructional presentation
contribute to its effectiveness? In this report, I address these
questions by proposing and applying an evaluative framework to five
online introductory calculus courses. The framework is informed by
cognitive research and research in mathematics education. As a
demonstration of the framework, I analyzed each course’s instruction
for the chain rule, a procedure that is mathematically interesting,
commonly encountered within and outside of calculus, and challenging
for the student. The application of this framework revealed that the
courses represented very different ideas on what it means to ``know''
calculus. In particular, the roles of rigor, exploration, computation,
and intuition in learning received varying amounts of attention.Carla van de Sande University of Pittsburgh carlacvds@gmail.com |

The
Relationship between Students’ Understanding of Functions in Cartesian
and Polar Coordinate Systems

The concept of function is first presented to students in the setting
of real numbers and the Cartesian plane. The broad definition of
function, as a relation where to every input corresponds only one
output, is complemented by representations specific to the rectangular
coordinate system, like the vertical line test. When students extend
their work with functions to other contexts, such as the polar
coordinate system, these particular representations often replace the
universal definition of function. Students rejected
as a function in polar coordinates because, as a circle, it “doesn’t
pass the vertical line test”. This study is a prelude to further work
on the concept of function in the multivariate context, and was
designed to measure how students deal with generalizations of the
function concept outside of the Cartesian coordinate setting, but still
in the single variable context.Draga
Vidakovic Georgia State University dvidakovic@gsu.edu |
Mariana
Montiel Georgia State University matmxm@langate.gsu.edu |

Tangül Kabael Anadolu Ünversity tuygur@anadolu.edu.tr |
Nikita Patterson Georgia State University matndp@langate.gsu.edu |

A Comparison
of Two Mathematicians’ Use of an Inquiry-Oriented Differential
Equations Curriculum

Orchestrating class discussions of mathematics in ways that conform to
reform-based principles of mathematics instruction can be challenging
to instructors accustomed to lecture-oriented classrooms. In
particular, instructors are challenged to direct classroom activities
and discussion in mathematically productive ways, while simultaneously
encouraging students to seek mathematical authority not in their
teacher, but in their own mathematical reasoning and judgment. In this
report, I investigate the experiences of two university mathematicians
during their first attempts to foster whole-class dialogue and
argumentation using an inquiry-oriented differential equations
curriculum. I argue that differences in class outcomes were associated
with the different ways each instructor negotiated the tension between
serving as a mathematical authority and enabling students to find
mathematical authority in legitimate mathematical reasoning. Balancing
these roles shapes opportunities for student learning, and
understanding such effects offers support to university mathematics
instructors who wish to adopt reform-based principles of instruction in
their classrooms. Joseph F. Wagner Xavier University wagner@xavier.edu |

Proving
Styles in Advanced Mathematics

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

Ian Whitacre San Diego State University ianwhitacre@yahoo.com |

Meeting the
Needs of the Client Disciplines: An Initial Look at Mathematical
Abilities of Students in Science, Technology, and Computer Information
Systems

There are calls for greater communication between mathematics
departments and the various client departments whose students require a
strong foundation in mathematics from organizations including the MAA
and Project Kaleidoscope (PKAL). Buffalo State, as part of a U.S.
Department of Education Title III grant, has formed a working group
comprised of mathematics, science, technology, and computer information
systems faculty whose objective is to gain understanding of the needs
of the various client departments, assess students’ abilities in
identified areas, and share the findings with departments. The
discussion of results will allow for building better understanding
among departments and generating ideas for the improvement of
instruction and student learning to be delivered through a series of
faculty workshops. The focus of this report is on the development and
administration of a draft assessment instrument. Analysis of initial
pilot results will be discussed along with implications for
undergraduate mathematics courses serving students in our partner
disciplines. David Wilson Buffalo State, SUNY wilsondc@buffalostate.edu |

Lesson
Planning Practices of Graduate Student Instructors in
Mathematics: Procedures, Resources and Planning Time

This report describes an empirical study of the lesson planning
practices of seven novice graduate student instructors (GSIs) teaching
a university precalculus course. We describe the salient features
of the processes that the GSIs typically followed when planning their
lessons, the amount of time spent on different parts of this process,
and the resources that the GSIs employed while crafting their
lessons. The GSIs in this study typically formulated objectives,
consulted resources, selected activities and (in some cases) attempted
to manage instructional time while planning. We found that GSIs
typically devoted less and less time to lesson planning during their
first semester of teaching, and that they typically use the resources
that were provided to them (prepared lesson plans and the course
textbook), although these were used a starting point for developing
more personalized lessons, rather than as exact “blueprints” for
instruction.Dale Winter University of Michigan amanitav@umich.edu |
Matthew
DeLong Taylor University mtdelong@tayloru.edu |

Sense of
Community: Contributing Factors and Benefits

In this qualitative study we explore how assessments contribute to
building a sense of community (SOC) in the classroom of an
undergraduate abstract algebra course. Strike (2004) describes
community as a process rather than a feeling and outlines four
characteristics of community: coherence, cohesion, care, and contact.
Coherence refers to a shared vision; cohesion is the sense of community
that results from the shared vision; care is a necessity to initiate
one into the vision, and contact refers to structural features of the
community. Using a grounded theory approach we analyzed student
interviews and report on the contributing factors to SOC as described
by students as well as perceived benefits by these students. We found
that contributing factors to the SOC align with Strike’s 4 C’s
definition of community and fall into two large categories: teacher and
environment. The contributing factors provide a model for a teacher
that wishes to build a SOC in his classroom, and the benefits provide
support for doing so.Nissa
Yestness University of Northern Colorado Nissa.Yestness@unco.edu |
Hortensia
Soto-Johnson University of Northern Colorado Hortensia.Soto@unco.edu |
Casey Dalton University of Northern Colorado Casey.Dalton@unco.edu |

Development,
Implementation and Assessment of Pre-Service Mathematics Course through
University/K-12 Partnerships in a NSF-funded Appalachian MSP

A partnership-driven approach to the creation, implementation and
assessment of a set of high quality mathematics courses in pre-service
teacher education is described within the framework of a comprehensive,
rural Mathematics and Science Partnership (MSP). The MSP course
development teams are mathematics disciplinary and education faculty at
nine Institutes of Higher Education (IHEs) and K-12 mathematics teacher
from 51 school districts in the central Appalachian regions of
Kentucky, Tennessee, and Virginia. Evaluations of sixteen mathematics
courses, the enrolled pre-service teachers and the teacher education
programs will be discussed relative to their impact on teacher
education programs in the participating IHEs. These include a
longitudinal study of the demographic and academic characteristics of
the pre-service teachers and the course impact on their cognitive,
affective and pedagogical domains. The sustained nature of the
partnership approach to improvement of content and teacher quality will
be highlighted in the presentation.John Yopp University of Kentucky jyopp@email.uky.edu |
Wimberly
Royster University of Kentucky royster@uky.edu |

Mathematical
Reasoning as Seen Through the Lens of Conceptual Blending

The purpose of this report is to describe how several seemingly
different phenomenon in mathematics education can be seen through one
unifying lens with the use of Fauconnier and Turner’s theory of
conceptual blending. This theory describes how humans reason and learn
by combining familiar mental spaces or frames into a new blended space
or an integrated network of blended spaces. I use three examples of
theoretical frameworks from the mathematics education literature,
including data analyzed within these frameworks, and show how they may
be reframed using conceptual blending. The examples are 1) the
notion of the key idea of a proof, 2) metaphorical and metonymic
relationships in a structured understanding of the concept of
derivative, 3) the emergent models heuristic from realistic mathematics
education. By explicating these three examples I hope to illustrate how
conceptual blending may play a role in our work in mathematics
education and to engage our community in a discussion of these
possibilities.Michelle Zandieh Arizona State University zandieh@asu.edu |