2006 Conference on Research in Undergraduate Mathematics Education
Contributed
Papers
and Preliminary Reports
Abstracts
Reasoning with Time as Parameter in Differential
Equations
Karen Allen karen.allen@valpo.edu 
This paper proposes a characterization of parametric reasoning: how students reason with the dynamic parameter time to support their mathematical activity and deepen their understandings of mathematical concepts. The research was conducted before, during, and after participation in instructional tasks and classroom discourse that focused on understanding and finding solutions to linear systems of differential equations. Students participated in the following identified mathematical practices which characterize parametric reasoning: using the idea that time never stops, reasoning simultaneously in a qualitative and quantitative manner, reasoning by moving from discrete to continuous imaging of time, and reasoning by imagining the motion. Examples of this reasoning are provided from two situated case studies in one classroom teaching experiment. Implications of this research include the possibility that instructional activities can build on student reasoning for students to understand and use the mathematics of change at the middle school, high school, and the university.
pbenko10@yahoo.com 
The learning of probability concepts is very complex (Piaget & Inhelder, 1975; Fischbein, 1975). Garfield and Ahlgren (1988) recommended that longitudinal studies be conducted. This study used qualitative methodology to analyze data in order to describe how a group of students build their probabilistic understanding from grades six, seven and twelve. This paper describes changes in students' representations of outcomes of dice games. I report on the verbal, written, and physical representations of student ideas developed during problem solving sessions. Students produced lists, written mathematical statements, pictures, and graphs. They used their representation to develop convincing solutions to problems posed. This study provides evidence that students did represent their ideas in a meaningful way to understand fundamental probabilistic concepts while working together under conditions of investigating rich tasks, collaborating and justifying their solutions, and having the opportunity to think deeply about their ideas before they receive formal instruction in probability.
Exploring
Calculus Students' Understanding of L'Hopital's
Rule
Bradley Beauchamp bkbeauc@ilstu.edu 
It is observed that students often have difficulties in understanding why L'Hopital's Rule works. These difficulties may stem from a shallow understanding of indeterminate forms, functional behavior, and rate of change. This study was designed after the conclusion of a preceding study concerning students' understanding and recognition of indeterminate forms. As such, this study attempts to describe a spectrum of students' conceptions of L'Hopital's Rule and to discover the depth of mathematical connections that are associated with those conceptions. The research questions, and interview protocol, are informed by APOS Theory. Data were collected in the form of written tasks in an interview setting from ten current Calculus II students. The tasks and interview questions were designed to develop a picture of the type of understanding students may have of L'Hopital's Rule and to what degree that understanding is connected to relevant topics in calculus.
On
Transcendental Numbers: Teachers' Understanding and Their Practices
Tanya Berezovski 
The miraculous powers of modern calculations are due to three inventions: the Arabic Notation, Decimal Fractions and Logarithms (Cajori, 1919). The mathematical concept of a logarithm plays a crucial role in advanced mathematics courses, including calculus, differential equations, number theory, and complex analysis. This report is a part of a larger study on the understanding of logarithms and issues involved in the development of this understanding. Specifically, I focus on inservice high school teachers' understanding of logarithms and how their understanding connects to their teaching of the concept.
Talking
Math: Mathematical Language and Explanations
in
a Course for Prospective Secondary Mathematics Teachers
Irene.bloom@asu.edu 
High quality mathematical discourse has been advocated for all levels of mathematics. The National Council of Teachers of Mathematics list “Communication” as a major content strand. NCTM also states that is up to the mathematics teachers “to initiate and orchestrate this kind of discourse and use it skillfully to foster student learning”. How do teachers learn to facilitate this kind of discourse when their ability to engage in high quality discourse may be minimal? This presentation reports findings from an ongoing project investigating mathematical knowledge and problem solving abilities in prospective high school mathematics teachers in the context of a capstone content course. In particular, I will characterize the nature of their mathematical explanations and verbal reasoning at the beginning of the intervention and document and describe the changes that occur as the instructor sets classroom expectations and facilitates mathematical discussions.
Students'
Reinvention of StraightLine Solutions
to
Systems of Linear Ordinary Differential Equations
Howard Blumenfeld

Chris Rasmussen chrisraz@sciences.sdsu.edu 
This paper analyzes how students who work in an inquiryoriented classroom think and develop solution methods for analytic solutions to systems of two linear ordinary differential equations (ODEs). In particular, we seek to understand how students construct and interpret straightline solutions (SLSs). SLSs are significant mathematical ideas because they serve as the basic building blocks for all other solutions. In particular, SLSs are eigensolutions that span the solution space. This report will also detail the role of proportional reasoning in the process of reinventing SLSs. Our theoretical orientation comes from a socioconstructivist view in which individual math activity is related to emerging classroom meanings and practices. Analysis of student work and justifications suggests that their mathematical understanding of SLSs is a dynamic and evolving entity that matures through classroom discussion and activity. Lastly, this study examines implications for teaching and for the revision of instructional materials.
An
inquiry into
Students’ Understanding of Linear Algebra
Marianna Bogomolny mbogomol@sfu.ca 
Students’ understanding of linear algebra and the effects of different teaching methods on students’ understanding have been investigated from a variety of perspectives. However, these questions have not been examined through the lens of studentgenerated examples. Since, in order to achieve understanding, students have to be engaged in a mathematical task, the present study investigates whether and in what way the examplegeneration tasks influence students’ understanding of linear algebra. In particular, the study examines students’ ability to construct examples for mathematical statements and objects in the undergraduate linear algebra course. It aims to analyze and describe what difficulties students encounter when constructing examples, and how examplegeneration tasks can inform researchers about students’ understanding of linear algebra.
Exploring
Mathematical Exploration:
How
College
Students Pose and Solve Their Own Problems in Classroom Setting
Jinfa Cai jcai@math.udel.edu 
In the past several years, we have started to investigate the processes that constitute the mathematical exploration processes of solvers. For example, we examined the mathematical explorations of two college students as they solved an openended computer simulation task that involved the path of a Billiard Ball. The results of this study suggest that mathematical exploration can be characterized as a recursive process in which solvers determine goals of action as they formulate their problems, solve the problems, and reflect upon their solution activities to formulate new problems. The results of the study contribute to the development of conceptual frameworks and research tools to capture mathematical exploration processes. A later study was designed to extend the findings of the earlier study by broadening the scope of the analysis to include new tasks and settings. This study illustrated how solvers’ cognitive actions can inform their evolving understanding of the task environment, which in turn motivates and enables them to seek out new problems to solve.
Students'
Conceptions of Indeterminate Forms
Jim Cottrill cottrilj@ohiodominican.edu 
Bradley Beauchamp bkbeauc@ilstu.edu 
We observe that students often have difficulties in understanding why L'Hopital's rule works, due possibly to a shallow understanding of indeterminate forms. This research wants to determine whether students recognize indeterminate forms simply through rote memorization or through rich mathematical connections. This study attempts to describe a spectrum of students' conceptions of indeterminate forms and discover what depth of mathematical connections is associated with those conceptions. The research questions are informed by APOS Theory, as was the interview protocol. Data were collected in the form of written tasks in an interview setting from six students in Calculus II. The results indicate students had difficulties due to poor notions of division of numbers. We need to delve into the student's ability to relate division to multiplication. We propose that instructors design activities to reinforce the definition of division of numbers. Specific suggestions in precalculus and Calculus I topics will be offered.
Intermediate
Algebra Students’ LanguageBased Knowledge
and
Their Conceptualization of Function
Hamide DoganDunlap hdogan@utep.edu 
I propose to discuss a group of intermediate algebra students’ conceptualization of the concept function when prior knowledge is dominated with common language meanings. Here, the term "common language knowledge/meaning” is used for the meanings that are used outside the context of mathematics. For instance, when “function” is used to describe the role of a teacher, it is considered that the common language meaning is intended. Two intermediate algebra students’ interviews and their responses on a set of posttest questions and definitions are used to argue that existing cognitive structures with nonmathematical origin may have implications for the learning of mathematical concepts.
The
Contrast of Hassan’s and Rich’s
Understandings
of
First Order Differential Equations:
‘Knowing
What To Do and Why’ and ‘Rules Without
Reasons’
John E. Donovan II john.donovan@maine.edu 
In this presentation results will be reported from a research project designed to investigate two questions: What is the nature of students’ understanding of algebraic and graphical representations of firstorder differential equations [FODEs], and what is the nature of students’ understanding of the connection between these representations and their solutions? A series of 4 interviews (1 background, 3 on FODEs) were designed to observe students’ thinking about FODEs within different problem contexts. This talk will focus on two of the students, Hassan and Rich, who came to understand differential equations in very different ways. Hassan had a well defined network of connections for FODEs in which different representations stood for the same thing, and the concept of differential equation was strongly connected with the notion of solution. For Rich, FODEs in and of them selves had very little meaning other than as inputs to procedures.
Student
Obstacles and Historical Obstacles to Foundational Concepts of Calculus
Rob Ely wambulus@gmail.com 
This study will report on the results from a questionnaire given to university calculus students in September 2005 and again in December, and informed by interviews administered in November. Students will be categorized by the epistemological obstacles they are experiencing to the foundational calculus concepts of limit, function, continuity, and the real number line. Briefly, an epistemological obstacle to a concept is a misconception that is necessarily, or commonly, a part of the concept. Potential epistemological obstacles are drawn from the research of several theories of undergraduate mathematical thinking, including mathematics as metaphor, APOS theory, prototype theory, and concept images. I will compare the combinations of obstacles the students display, and how these combinations change over time, with the combinations and progressions of obstacles found in the historical development of calculus. This should help to illuminate the relationship between the historical structure(s) and the cognitive structure(s) of calculus.
Teaching
Innovations for Related Rates Problems in First Semester Calculus
Nicole Engelke 
Related rates problems in first semester calculus are a source of difficulty for many students. These problems require students to be able to visualize the problem situation, attend to the nature of the changing quantities, reconceptualize the variables in a geometric formula as functions of time, and relate these functions either parametrically or through function composition. Being able to successfully solve a related rates problem by engaging in the problem solving behaviors described above relies on the students’ ability to engage in transformational/covariational reasoning. I have developed a sequence of teaching activities that employs a computer program designed to foster the students’ exploration of related rates problems in a covariational context. I am investigating the impact of these activities on students’ abilities to solve related rates problems.
The
Calculus Concept Inventory
Jerome Epstein jepstein@poly.edu 
This is a report on the development and validation of a diagnostic instrument, The Calculus Concept Inventory (CCI). It is patterned on the Force Concept Inventory (FCI) in physics, which measures conceptual understanding of the most fundamental concepts of mechanics. The “normalized gain” is widely shown as independent of the entry level of students, of instructor and textbook, but strongly dependent on teaching methodology. It is crucial to determine whether this holds in math as well. The hope to help resolve the “math wars” is to develop a valid and accepted test that can distinguish between methodologies. We report on the process of development, results on initial pilot and first round of field testing, and statistical evaluation of results.
We
Know What They
Knew, but What do They Know?
The timfc@wam.umd.edu 
Steven Jones The themuppies@hotmail.com 
In our presentation, we propose to describe the daily class activities of a nontraditional abstract algebra class. The description will include both a general description and thick description that was created by observation and video data. As our second contribution to the field, we will offer remarks about the types of mathematical understandings, proficiencies, and habits that the students developed while part of this course. Finally, we will offer a brief report on some these students’ current understandings of group and ring theoretic material (approximately six months after their last course meeting), drawn from a brief written instrument and diagnostic interview.
Students
Ideas on Functions of Two Variables
Maria Trigueros Gaisman Instituto Tecnológico Autónomo de Mexico trigue@itam.mx 
Rafael MartínezPlanell rafael@math.uprm.edu 
The notion of a multivariable function is of fundamental importance in mathematics and its applications. This is a report of a study that uses APOS theory as theoretical framework to study students’ learning about functions of two variables. A very brief introduction to the basics of APOS theory will be presented. It will be followed by a particular genetic decomposition for functions of two variables. An instrument was designed to test the different components of the genetic decomposition. A small group of students was chosen and interviewed using the instrument as a basis. The results were used to improve the instrument and after revision it was then applied to another group of students who were also interviewed. Results from the student interviews show that they have difficulties with the basic aspects of functions of two variables. The results will be discussed, particularly as they relate to the proposed genetic decomposition.
Dialogue
as a Practical Tool for Writing Proof
Soheila Gholamazad 
This research examines students’ engagement in creating a dialogue as a means towards understanding and creating proofs. In this study I adopt the communicational approach to cognition based on the work of Sfard, according to which thinking is a special case of activity of communication. Participants in this study were preservice elementary school teachers. I encouraged students to write down the dialogue that they have with/or among themselves while they are thinking and trying to understand or create a proof. The results showed the method of proving through writing dialogue would be practical heuristic for involving students in the process of understanding and creating a mathematical proof.
Conceptions
held
by (future) primary school teachers
concerning
the concept of angle
Alexandra Gomes magomes@iec.uminho.pt 
In the process of creating mathematical knowledge, concepts perform a crucial role, being one of the foundations of the mathematical building. Their meaning has to be constructed by the individual in interaction with different contexts, based either on experience or on abstract references. Therefore, definitions are essential in mathematical activity since they constitute a basic component of mathematical knowledge. On the other hand, representations are a fundamental link in the codification and understanding of mathematical concepts. This study will explore the conceptions that (future) primary school teachers have of a basic elementary geometrical concept (angle), especially concerning the role played by definitions and representations.
Network
Analysis as
for
Understanding Graduate Student Communities
Timothy Gutmann

Eric Hsu 
Emiliano Gomez emgomez@Math.Berkeley.edu 

Recently professional developers and mathematics education researchers have begun to focus on the needs of mathematics graduate students. This proposal discusses the shortcomings of Lave and Wenger's community of practice model and Nodding's caringbond model as theories to describe learning within a graduate student community. It proposes social network analysis as a way to empirically generate a working theory of community learning.
Patterns
of Change in the Teaching Actions and Behaviors of an InService Teacher
Guershon Harel 
Osvaldo Soto osoto@ucsd.edu 
The case study reported here examines the teaching practices of one inservice teacher who participated in an onsite professional development (PD) for two years. In this report, we characterize patterns of change in the teacher’s teaching practice over that two year period. Our analysis, focuses on five major categories of change in the teacher’s teaching practices; changes in the way the teacher solicits student ideas, in how she deals with student errors, in capitalizing on solutions offered by students, and in the type of questions she chooses to pose.
Shandy Hauk hauk@unco.edu 
Jennifer Kreps 
April Brown Judd Midcontinent Research for Education and Learning april@thejudds.org 
Rhoda Deon rhodadeon@yahoo.com 
The article provides results of a study of the pedagogical content knowledge (PCK) of middle school mathematics teachers enrolled in a professional development program. About onethird were already "Highly Qualified" according to No Child Left Behind Act of 2001 criteria and twothirds were not. Measures of PCK included subscores on content, syntactic, anticipatory, and classroom action knowledge from a taskbased interview and on a written test based on national standardized tests for teachers. Results indicate that a truly “Highly Qualified” teacher, one with robust PCK, may benefit most from professional development aimed specifically at building and developing selfawareness of mathematically rich anticipatory and classroom action knowledge in addition to opportunities to enrich mathematical content understandings.
Engineering
as Bridging Undergraduate and K12 Mathematics
in
the NSF Math and Science Partnerships
Margret A. Hjalmarson 
This study presents an analysis of K12 and undergraduate engineering education initiatives within an NSF grant program. The implications for bridging gaps between K12 and undergraduate include mathematics and engineering content and curriculum since mathematics courses are a gateway to engineering programs and engineering contexts provide natural venues for mathematics investigations. The types of initiatives include faculty development, K12 teacher development, curriculum design, and projects designed for recruitment and retention of underrepresented groups in engineering.
Patterns
of Motivation and Beliefs among BeforePrecalculus
College
Mathematics Learners
Pete Johnson Eastern johnsonp@easternct.edu 
This study focused on college mathematics students’ motivation to learn and their beliefs about the nature of mathematics, and on the linkages between these two constructs. Based on the selfdetermination theory of Deci and Ryan (1985), the literature on belief systems (Green, 1971; Rokeach, 1968), and on reviews of research, five aspects of motivation to learn and six common beliefs about mathematics were considered in this study. Using data from interviews, questionnaires, videotapes, and other documents, three different profiles emerged among the eight students in the study. These profiles were termed “conceptually motivated,” “externally motivated,” and “future value motivated.” Further research is needed to determine the prevalence of these motivational profiles in other college mathematics settings.
What
LearnerGenerated Examples Reveals about Students’
Understanding
of Combinatorial Structures?
Shabnam Kavousian Langara College/Simon Fraser University skavousi@langara.bc.ca 
This study investigates students’ understanding of combinatorial structures, and examines their difficulties through an example generation task. The participants are undergraduate students enrolled in a mathematics course for liberal arts students. The participants were presented with a task in which they were asked to generate an example of a combinatorial problem given a presented combinatorial structure. Their responses were analyzed to examine their difficulties in understanding specific combinatorial structures. As a result of this study, I will suggest some practicable ways to help learners acquire better understanding of this topic.
Mathematical
Justification, Parallel Constructions of Knowledge and the
Instrumentation Process
Ivy Kidron ivy@jct.ac.il 
tommyd@post.tau.ac.il 
Processes of knowledge construction about bifurcations of dynamic processes are investigated on the basis of the epistemic actions model for abstraction in context (RBC). The subject is an experienced learner of mathematics. The motivation of achieving a justification for finding the bifurcation points drives her entire learning process. A previous study (Kidron & Dreyfus, 2004) demonstrated that the constructing actions may go on in parallel and interact. The interactions include branching of a new constructing action from an ongoing one, combining or recombining of constructing actions, and interruption and resumption of constructing actions. The analysis showed that the process of justification is related to parallel constructions and in particular, the combining of constructions leads to enlightenment, not in the sense of a formal proof of the statement we want to justify but as an insight into the understanding of the statement. This paper also makes a theoretical contribution to the development of the theory of the RBC model. Specifically, we have analyzed in depth the influence of a particular component of the context, namely an instrument. More specifically in the case at hand, the instrument was a computer tool, Mathematica, and the influence was on the interacting parallel constructions, branching and combining.
Expanding
the Participation Metaphor:
What
is the Development
of Mathematical Practice?
Karen D. King Karen.d.king@nyu.edu 
This theoretical paper explores the extension of
the
participation metaphor as outlined by Sfard
(1998) to
research on undergraduate mathematics education. In particular, by
building on
the work of
A
Framework to Examine Definition use in Proof
Jessica Knapp 
Mathematicians view a definition as having very specific characteristics. Besides defining mathematical objects, definitions give both structure to a proof in a global sense as well as warrants to logical implications in a local argument. Thus definitions play a central role in many proving tasks. Students are required to be able to create and interact with proofs and definitions in a variety of ways. Students are expected not only to produce proofs in homework, but also textbooks and lectures are written with the expectation that students can read and understand proofs and formal definitions. In this talk I propose a framework with which to examine students’ uses of definitions and theorems in proving. The framework is a synthesis of previous literature and research results. I will also seek to illustrate the aspects of the framework using student data from a workshop in advanced calculus.
A
Tale of Two Backgrounds: Snapshots of Graduate Students’
Knowledge
of
Student Thinking in Calculus
David Kung St. Mary’s 
Natasha Speer 
Improving teachers’ knowledge of student thinking appears to be a successful model for K12 professional development. For graduate students, with little or no preparation for teaching, opportunities to learn about students’ thinking come primarily onthejob. We examined knowledge gained by two different graduate student populations early in their teaching careers. While both groups interacted with students during office hours and graded homework and exams, their inclass teaching differed significantly. Some taught traditional recitation sections, mostly answering student questions. Others facilitated Emerging Scholars workshops, writing challenging calculus problems, observing as students work in collaborative groups, and providing Socratic help when necessary. Using a framework adapted from Cognitively Guided Instruction, these “snapshots” show the former group has minimal knowledge of student thinking about key topics in calculus while the latter group has welldeveloped knowledge of student solution strategies, difficulties, and typical misconceptions.
Impact
of Enhancing Reflection and Discourse in College Algebra
Christine Larson chjlarso@indiana.edu 
Robert Mayes rmayes@math.wvu.edu 
The learning of many postsecondary mathematics
students is impeded
by their failure to (1) adopt the vocabulary and notations commonly
accepted by
the mathematics community and (2) recognize fundamental concepts
underlying
course content. The intent behind constructing a learning environment
where
students reflect on vocabulary, notations, and concepts and argue about
what
these things mean and why they are important is to provide the
scaffolding for
an apprenticeship based model of learning, thus facilitating greater
student
engagement and learning. This study aims to address the question: Can
structurally changing undergraduate math courses to include student
reflection,
discourse, and inquiry in routine classroom practice improve student
learning?
The study will target a “high risk” group of students enrolled in the
applied
algebra course at
The
Examination of Factors Influencing the Integration of Computer
Algebra
Systems in
UniversityLevel Mathematics Education
Zsolt Lavicza zl221@cam.ac.uk 
Accumulated evidence indicates that the integration of technology into mathematics education is slow compared to the predicted quick spread two decades ago. Schoollevel studies demonstrated that technology integration is highly dependent on teachers' conceptions of technologyassisted teaching, yet, researchers rarely investigate this topic at universities. During the past year, I conducted an exploratory study to examine mathematicians' conceptions of CASassisted teaching. I interviewed mathematicians, observed classes, and collected course material at a variety of universities in three countries. In my talk, I plan to share the findings of this study and to highlight the differences between mathematicians' and school teachers' CASrelated conceptions, as well as to discuss the variations in mathematicians' thinking about CAS in the participating countries. In addition, I am developing a largescale quantitative study to examine CAS integration in university mathematics education programs. Finally, I intend to share and to discuss my preliminary questionnaire design and sampling strategy with the audience.
The
Evaluation of the Effectiveness of preFreshman Summer Mathematics
Courses
on
the Academic Performance of Underrepresented Minority Engineering
Students
Zsolt Lavicza zl221@cam.ac.uk 
Darryl Koch koch@umich.edu 
Universities in the Unites States report that it is becoming increasingly difficult to attract sufficient numbers of science and engineering students. Moreover, recruiting and retaining minority students is even a more challenging task. Thus, a number of programs have been established to support minority students before beginning their freshman year. In many programs, adequate mathematics preparedness has been identified as a critical factor for minority students. future success. At our university, various prefreshman summer programs have been serving minority engineering students for more than 20 years. Our studies indicate that participants in these programs have better graduation rates than their counterparts who do not participate, but little is known about the specific effect of these programs on students. mathematical preparedness. Therefore, we decided to develop a study to examine the effect of prefreshman summer math courses on students. mathematical preparedness and to assist us to improve these programs. In our talk, we shall outline our research approach and the initial results of our investigation.
Connecting
Tasks in PreService Teacher Mathematics Education
Roza Leikin rozal@construct.haifa.ac.il 
This study explores employing connecting tasks in mathematics education of preservice mathematics teachers (PMTs). This report presents ongoing study with PMTs, which is based on the findings of our longitudinal study with inservice mathematics teachers. We found that inservice teachers' mathematical and pedagogical knowledge is curricular oriented and mostly craft and prescribed. We found that teachers' rarely remind systematic sources of knowledge associated with connecting tasks. In this research project we explore development of PMTs' knowledge in systematic mode. The research is performed in the context of geometry course focusing problem solving in different ways. It employs observations as well as written questionnaires. The 3D model of teacher knowledge is applied for the analysis of PMTs' knowledge development.
Building
an Understanding of Student Use of Graphing Calculators
as
a Tool for Problem Solving
Allison McCulloch amccullo@eden.rutgers.edu 
Graphing calculator use plays a significant role in calculus instruction and assessment. The purpose of this study is to begin to construct an understanding of how calculus students use the graphing calculator as a tool when they are working independently and how they feel its use influences their problem solving experiences. Fourteen students participated in a taskbased interview. Four of these students participated in a followup interview during which they reflected on the tasks and their feelings about calculator use. The results of this study show that students are often triggered to use the calculator to find a new approach for solving a problem. This new approach is often a visual one. Additionally, there were sex differences in student assignments of authority.
A
Continued Look at the Impact of WebBased Linear Algebra Tools
on
Understanding
David E. Meel 
This talk will outline a variety of webbased linear algebra tools which focus on concepts such as linear transformations, eigenvalues and eigenvectors, change of bases, matrix multiplication, diagonalization, etc. By utilizing a variety of data collection methods, such as journal entries, written responses, concept mapping activities and interview questions, the talk will discuss how the students interacted with the webbased tools and what connections they built. Indications point to changes in student descriptions and connections they made as a result of interacting with the webbased tools. Samples of student work and statements concerning the use of such tools will be presented.
Is
this the answer? Is this what I was supposed to do?
Control
Structures
in Introductory Calculus Textbooks
Vilma Mesa vmesa@umich.edu. 
Preliminary report of an analysis of 9 introductory calculus textbooks that looked at strategies available to students for (1) determining whether an action was relevant when solving a problem and (2) deciding that the problem was solved. This is known as the control structure, and it plays an important role in using validation strategies in mathematics. The books differ in terms of audience (Honors, Regular, Applied, Science & Engineering) and in terms of influence by the calculus reform movement. The study describes the control strategies that are present in lessons and examples associated with the derivative (~800 examples) and establishes differences across textbooks. There are two main hypotheses in this study: (1) Strategies tend to rely more on content at stake in books intended for honors students, and (2) Reform oriented textbooks provide the control structure more explicitly. Results have implications for research, teaching assistant training, and teaching.
Learning
What Needs to be Justified in Mathematical Proof
Susan Nickerson snickers@sunstroke.sdsu.edu 
Chris Rasmussen chrisraz@sciences.sdsu.edu 
The overall goal of this preliminary report is to investigate undergraduate students' epistemological shifts in their mathematical justification and to identify pedagogical factors that foster and/or constrain critical shifts in students' ability to create mathematical arguments. We view proof as a social process in which participants of a learning community move from peripheral participation to more full members of a community engaged in the mathematical activity of proving. This report analyzes the impact on students of an instructional intervention that modifies of the classical format of a twocolumn proof which reifies the presentation of a proof and the questions asked by the writer or a reader of the proof.
Model
Analysis in Undergraduate Mathematics Education Research
Michael Oehrtman 
This talk will outline the data analysis technique of model analysis theory and its potential applications to mathematics education research. Pioneered in physics education, this theory uses basic principles of quantum mechanics as an inspiration for accounting for nondeterministic, multistate student understandings. We apply the techniques to two conceptual areas in mathematics education, understanding of functions and limits. Model analysis of 76 secondary mathematics and science teachers participating in a professional development focusing on function concepts reveals little change in the model structures related to action and process views. Analysis of 42 students in standard undergraduate calculus courses reveals a slightly increased tendency of students to apply a wider variety of models, including incorrect models, at the end of the course. Students in an experimental course designed to foster particular conceptual models for limits, exhibited increased tendencies to choose these models and decreased tendencies to choose incorrect versions of them.
Considering
the Norms and Practices of Mathematicians
when
Analyzing an Advanced Undergraduate Mathematics Classroom
Katrina PiatekJimenez

Just as social norms, sociomathematical norms, and mathematical practices develop within a classroom community (Cobb & Yackel, 1996), such norms and practices have also developed within the community of mathematicians. These norms and practices influence the norms and practices within a classroom through the instructor, textbooks, and curriculum. As a result, when analyzing an advanced undergraduate mathematics classroom, it is important to consider the role that the norms and practices of the mathematics community play in the development of those in the classroom. During this talk I will present a proposed framework based on the Emergent Perspective that will also incorporate the influence of the community of mathematicians into the study of undergraduate mathematics classrooms. This framework is grounded in qualitative data collected during the study of two “Introduction to Proofs” classes. Data from this study will also be discussed in the context of the framework.
Developing
Deep, Connected, and Durable Understanding of Combinatorics
Arthur B. Powell abpowell@andromeda.rutgers.edu 
This study, part of a longitudinal investigation into the development of students’ mathematical ideas and reasoning, examines the deep, connected and durable understanding of combinations evidenced in students’ reasoning through transitivity to establish an isomorphism between contextually different problems. A goal of teaching combinatorics is to have students understand computational techniques of combinatorics and combinatorial notations as well as to use flexibly these techniques and notations to solve varied problems. However, mathematics educators are uncertain about how to achieve this goal. Moreover, the question is underresearched. The results of this study suggest that learners require time to enable ideas to dawn and mature, that problems that invite students to think deeply and discursively enhance their mathematical understanding, and that strand of varied, spiraling problems help students develop deep, connected, and durable understanding and facility with basic combinatorial ideas.
Effects
of DNRBased Instruction on the Teaching Behaviors of Algebra Teachers
Jeffrey M. Rabin jrabin@ucsd.edu 
Guershon Harel gharel@ucsd.edu 
We report further results from an ongoing study of changes in the teaching behaviors and knowledge base of algebra teachers following professional development institutes embodying "DNRbased instruction", a conceptual framework for the learning and teaching of mathematics. Teaching behaviors of two teachers are categorized and evaluated for alignment with DNR. Probable effects of these behaviors on student learning are conjectured. We document changes in teachers' practice following their experience of DNRbased instruction. The choice and effectiveness of teaching behaviors depend on adequate mathematical and pedagogical content knowledge.
Videocases as a
Didactical Intervention in an
"Introduction to Proof" Course
Manya Raman mjraman@rci.rutgers.edu 
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 bridgetype proof course at the university level. Students that had particular difficulty writing proofs were asked to volunteer for an extra help session. In this session, students watched and discussed videos of "experts" talking aloud while proving several of the proofs that were difficult for the students. The goal of the sessions was to help students connect informal and formal aspects of proof. In this talk, we discuss the relative advantages and disadvantages of this intervention, and we solicit ideas for improvement for the second iteration.
Students’
Proofs for the Shapes of Graphs of Solutions in the Phase Plane
Chris Rasmussen San Diego State Univeristy chrisraz@sciences.sdsu.edu 
Jill Nelipovich 
Bohdan Rhodehamel 

This paper reports on student proofs for the shapes of graphs of solutions in the phase plane for systems of two linear homogenous differential equations with constant coefficients. We define proof to be an argument and we draw on Toulmin’s scheme for the anatomy of an argument. Data for this analysis comes from an eightweek teaching experiment conducted in an undergraduate differential equations course in a large southwestern university. We found that students’ proofs relied on six sources (data) for the basis of their claims. In our presentation we will illustrate and clarify the students’ deductive arguments for solution graph shapes. Findings from our research contribute to contemporary characterizations of proof as a social process. The presentation will conclude with implications for teaching and instructional design.
Association
of Students' Conception of Limits with Their Levels of Reversibility in
the
Context of Sequences
Kyeong Hah Roh khroh@math.asu.edu 
This study explored how students conceptualize the relation between ε and N in the definition of the limit of a sequence. Students participated in the ε–strip activity which was specially developed to describe the relation between ε and N. This study results that students' understanding of the relation between ε and N fell into 5 major levels. In the classification of levels, it was regarded as an important factor whether or not students understand the arbitrary choice of ε, the dynamic feature of ε to decrease to 0, and the dependency of N on ε. It was also found that students’ levels of reversibility were associated with their understanding of the limit of a sequence. Actually, throughout the ε–strip activity, students’ reversibility and/or their conception of limit had been improved. It implies the development of reversibility should be considered in teaching limits of sequences.
Student
Understanding of Accumulation and Riemann Sums

Michael Oehrtman 
Student understanding of the conceptual structure of the Riemann sum definition of a definite integral is a topic on which little research has been done. This research uses 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 accumulation in college calculus students. Data shows that students with a strong understanding of Riemann sums are much more successful at setting up nontrivial definite integrals and solving nontrivial accumulation problems.
Students’
Strategies for Constructing Mathematical Proofs
in
a ProblemBased Undergraduate Course
Jennifer Christian Smith The jenn.smith@mail.utexas.edu 
Sera Yoo The 
In order to more closely examine the strategies and processes students employ when constructing mathematical proofs, we followed six students enrolled in a problembased undergraduate number theory course. During a series of interviews, the students were asked to construct proofs of various number theory statements. The students’ strategies for constructing proofs varied depending on context, but in each case they were primarily engaged in making sense of the mathematics, rather than applying a previously known proof strategy. In addition, their strategies varied greatly from one proof to another; they did not appear to show a preference for a particular approach to constructing a proof. We claim that the problembased structure of the course facilitated the development of these students’ relatively fluid approaches to proof construction.
Undergraduate
Mathematics Teachers' Knowledge of Students' Strategies
and
Difficulties with Derivative Problems
Natasha Speer nmspeer@msu.edu 
Sharon Strickland strick40@msu.edu 
Nicole Johnson john1968@msu.edu 
Beste Gucler guclerbe@msu.edu 
Teachers’ knowledge shapes decisions they make while planning instruction and while teaching. Knowledge of student thinking (e.g., typical solution paths, difficulties) appears to be an especially powerful influence on teachers’ practices and a productive site for professional development. These issues, however, have not been examined in many content domains. This research examined college mathematics teachers’ knowledge of student thinking about derivative and concepts that interact with learning derivative (e.g., function, limit). We analyzed participants' knowledge of student strategies and difficulties for tasks previously used in research on student thinking. Comparisons are made between participants' knowledge of student thinking and findings from research on student thinking. Analyses indicate that participants did not possess rich knowledge of student thinking. Specifically, most struggled to generate solution paths other than the ones they had used and anticipated only procedural difficulties. Implications for practice and design of professional development are discussed.
Students'
Use of Representations to Develop Ideas in Combinatorics
Lynn D. Tarlow ltarlow@ccny.cuny.edu 
This study documents the mathematical development
of a group
of students who built representations to solve challenging combinatorics
tasks and then refined and linked those representations in order to
develop an
understanding of the relationship between the tasks, the combinatorial
idea of
(m choose n), and Pascal's Triangle. The students' use of
representations was
critical for their development of an understanding of combinatorics,
a topic that has been noted to cause difficulty for college students.
This
suggests that ideas of combinatorial notation, perhaps, should not be
imposed
on students, but rather students should be given time to develop these
ideas in
environments that encourage them to recall, produce, refine, and
connect
representations.
Dialogue Based Teaching Practice in an
Undergraduate Analysis Course
Stine Timmermann Roskilde University, Denmark stinet@ruc.dk 
This is a report of the results of a pilot study of an undergraduate
analysis course at a Danish University, characterised as a reform
university. The specific course was examined as a part of a larger
project concerning the establishment of plausible connections between
teaching practices and students’ problem solving strategies and
reasoning. Data was collected in the form of classroom observations
based on a predeveloped schedule, semistructured interviews with
students and teacher and monitored problem solving sessions with
students grouped in pairs. The results reveal a discrepancy between the
expressed needs of the students (highly influenced by their
disabilities to solve the assigned tasks), the teacher’s intentions and
the actual behaviour of students and teacher in class. Possible
explanations for the course of events and guidelines for further
improvements are proposed.
Building
Mathematical Connections through Communications
Elizabeth B. Uptegrove uptegrovee@felician.edu 
Carolyn A. Maher 
We describe here a problemsolving session where, through collaboration and discussion, a group of students strengthened their understanding of some features of Pascal's Triangle by identifying connections between two isomorphic combinatorics problems with different surface features. The question that guides our analysis was: How did communication among students in a problemsolving group contribute to the group's success in making sense of problems in combinatorics? We use data from a longitudinal study that has followed a group of public school students from first grade through college. Data for this analysis are taken from an afterschool probleminvestigation session involving four high school students. The theoretical framework is provided by Maher and Sfard, who stress the importance of communication in mathematical thought. Our view, supported by our analysis of the students' conversation, is that discussing mathematical ideas and communicating about them are important prerequisites for recognizing connections between problems of equivalent structure.
Spontaneous
Online Discussions in Mathematics
Carla van de Sande
cav10@pitt.edu 
Spontaneous online discussion boards (ODBs) are forums where students post questions, receive explanations, and discuss both conceptual and procedural mathematical issues. In contrast to ODBs affiliated with a course or institution, spontaneous ODBs involve anonymous participants who do not have contact outside of their online discussions and who participate in discussions that are generated by queries originating from a wide variety of contexts. Spontaneous ODBs may be one of the few resources outside of school available to upper level mathematics students and appear to be a popular source of tutoring assistance. Although tutoring has been studied in other contexts (e.g. facetoface) and in courseinitiated ODBs, it has not been studied in the context of spontaneous ODBs. This research investigates tutoring exchanges in spontaneous ODBs on advanced mathematical topics such as limit and seeks to characterize the associated discourse community in terms of pragmatics and practices of instructional explanation.
StudenttoStudent
Questioning: A Case for Autonomous Scaffolding
Janet G. Walter jwalter@mathed.byu.edu 
This paper examines studenttostudent questioning and problem posing within a framework of personal agency, purposeful choice and learnerinitiated inquiry in problem solving. This qualitative study is part of a precalculus strand in a larger longitudinal study of students’ mathematical thinking. Six highschool students worked collaboratively on a task that suggested the use of polar coordinates, a topic with which these students were unfamiliar. Questions emerged as essential features of students’ problemsolving activities. Students asked questions that were linguistic representations of mathematical thinking and purposed toward making sense of a problem situation. Implications for teaching practice will be discussed.
How
do Undergraduates Complete Homework in Abstract Algebra
and
What Do They Learn From Doing It?
Keith Weber khweber@rci.rutgers.edu 
Three undergraduates were videotaped thinking aloud while completing homework assignments for two one hour sessions. Afterwards, they were asked what they learned by completing these exercises as well as more general questions about their strategies for completing their homework and their learning in general. The results of this study reveal the different strategies undergraduates use to complete their homework and how these strategies significantly affect what they learn from doing their homework.
Developing
Undergraduates' Understanding of Trigonometric Functions
in
a College Geomtery Course
Keith Weber khweber@rci.rutgers.edu 
In this talk, I will define what it means to understand a trigonometric function, using Gray and Tall's (1994) notion of procept. I will describe instruction based on this theoretical analysis. I will then report the results of a comparison study
Reconceptualizing
Mathematical Objects as
Mediating Discursive Metaphors
Aaron Weinberg aweinberg@ithaca.edu 
There are several frameworks used to describe students’ conceptions of mathematical entities as processes and objects from a Piagetian perspective. The work of Vygotsky and Bakhtin suggests focusing on the use of tools such as symbols and language that mediate thinking and discourse. Using these ideas, we present a new framework for describing functions as mathematical objects in discourse. Incorporating the ideas of facet, objectmediated discourse and a mediational toolkit, this framework is used to analyze undergraduate calculus students' discourse about functions. The components of the framework describe the metaphors students use to talk about and work with functions along with the focus of their discourse. The two framework components allow us to analyze different aspects of discursive mathematical objects, and suggest an instructional focus on developing discursive competencies as students participate in mathematical activity.
Effects
of Highly Contextualized Learning Activities on Students’
Mathematical
and
nonMathematical Recall
Dale Winter amanitav@umich.edu 
Many claims have been made concerning the
advantages of
embedding mathematical problems in “real world” contexts. One specific
claim is
that learning mathematics by solving such problems may enhance
students’ recall
of mathematical concepts and procedures (Blumberg et al., 2005;
Learner Generated Examples: From Pedagogical
Tool to
Research Tool
Rina Zazkis Zazkis@sfu.ca 
Watson and Mason (2005) make a compelling argument for the use of learner generated examples as an instructional tool. I wish to explore whether and how learner generated examples can be used as a research tool, that is, as a lens for researcher to interpret students' knowledge or understanding of subject matter. I will consider series of learner generated examples and invite inferences about the students. Further, among different perceptions on learning I will examine a possible view of learning as "enriching individual's example space".