Proceedings
for the Twelfth
Special Interest Group of the Mathematical Association of America
on Research
in
Undergraduate Mathematics Education
Conference on Research
in
Undergraduate
Mathematics Education
Marriott Raleigh City Center  Raleigh, North Carolina
February 26  March 1, 2009
(Click on the title of the presentation to download the corresponding proceedings paper)
BEST PAPER
eProofs: Student Experience of Online Resources to Aid Understanding of Mathematical Proofs
Lara Alcock Loughborough University (UK) l.j.alcock@lboro.ac.uk 
In this preliminary report I will discuss the development and testing of a set of eLearning resources termed eProofs. These eProofs have been developed with the aim of helping students to make sense of eight of the proofs in an Analysis course. They are available online, and each allows students to review the development of the proof, to watch a linebyline explanation of the reasoning used, and to see a further breakdown of the proof into largerscale structural sections (all with audio commentary). In the presentation I will demonstrate the eProofs and present 1) weekbyweek quantitative data on student use of this resource during the semester and revision period, and 2) qualitative student feedback on the perceived utility of the e Proofs. I will also welcome discussion about plans for future development of, and research about, these resources.
Workbooks for Independent Study of Group Theory Proofs
Gavin Brown
University of Kent (UK)

This report will describe a smallscale research project investigating students’ experience of workbooks designed to support independent study of proofs in a first course on group theory. We will set this description against research and theory on students’ learning of group theory and on epistemological beliefs and study habits in higher education. We will then present two types of data: 1) quantitative feedback from the whole class on the learning experience and practical matters such as time spent, and 2) qualitative analysis of observations and interviews with volunteer students who allowed their first hour’s work on a workbook to be recorded. Finally we will discuss our data in terms of general issues it raises for the design of tasks to support independent study of proofs.
Critical Experiences in Discussions Regarding Teaching: The case of graduate mathematics teaching assistants
Jason Belnap
Brigham Young University

Peer discourse regarding teaching is central to many contemporary forms of professional development (PD). Program success relies in part on participants abilities to contribute substantively to these discussions. What empowers participants to participate in these conversations?
A recent qualitative study of discussions among GMTAs in one PD setting revealed although they drew upon a variety of elements, their discussions were characterized by being
highly based on personal educational experiences (i.e. classroom experiences as students, as teachers, and as observers; and tutoring experiences as tutor and tutee).
To understand what experiences empower participants to contribute to discussions on teaching, we conducted a detailed analysis of transcripts from several sessions of one PD
program. We found that each type of experience differed in its crosssection of five major characteristics; these differences represent potential biases, strengths, and weaknesses
of each experience type. Details and implications for PD will be discussed in this paper/presentation.
Making Sense of Group Discussions: Analyzing discourse through content threads
Jason Belnap
Brigham Young University

Researchers study discourse to understand and improve discussions naturally occurring in contemporary forms of classroom activities and professional development programs. These discussions often have complex structures, posing signifcant analytical challenges.
During recent qualitative studies, we overcame many problems by applying principles and frameworks from social linguistics, breaking discussions into blocks (sequences) and developing the Framework for Conceptualized Function (FCF), which describes individual contributions function in building the a sequences total content. Imposing this block structure, however, makes analysis complex and accurately describing the structure problematic.
Through further analysis, we are nding that the FCF alone can describe the content structure of a discussion, producing not a block but a threadbased model. This approach simplifies analysis and more accurately describes how individual contributions build during a discussion. In this presentation, we describe and contrast these two approaches. We invite participants to provide insights and directions for this line of inquiry.
Innovative methodologies: the study of preservice secondary mathematics teachers’ knowledge
Tetyana Berezovski Saint Joseph's University tberezov@sju.edu 
This study is a contribution to the ongoing research in mathematics teachers’ education. It focuses on designed tasks as a methodology to probe teachers’ knowledge for teaching. It is guided by the belief that teachers’ knowledge required for teaching is very complex and multifaceted. The study introduces two research designed tasks as an effective data collection tool to investigate preservice teachers’ knowledge of mathematical concepts. The tasks designed for this study proved to be efficient not only as research tools but also as effective learning environments. They provide learners with an opportunity to engage in mathematical activity and contain an element of surprise that requires preservice teachers to think ‘on their feet’.
An Experiment in Teaching Algebra to Teacher Candidates
Ann D. Bingham
Peace College

Carolann Wade
Peace College & Wake County Public School System

Our goal in this project is to introduce elementary ed/special ed majors to a constructionist pedagogy that is shown to produce mathematics students who can think and work through mathematical problems and know that doing mathematics is not just a formulaic exercise. Our project involves assessing the preservice teachers’ understanding of algebra. We then plan to devote some class time to teaching algebra content in an active learning environment. We hope to find that this increases their understanding of the mathematics and that the teacher candidates see the benefit of this pedagogy in the math classroom. Our assessments will include pretests and posttests on the content knowledge, written responses asking students to reflect on the pedagogical methods used in their education and in the special classes and interviews with selected students.
Precalculus Student Understandings of Function Composition
Stacey A. Bowling Arizona State University stacey.bowling@asu.edu 
Few published studies have focused explicitly on students’ understandings of function composition and the mental imagery that students construct when attempting to solve problems that involve function composition. The present study addresses this research gap, studying the mental processes involved in understanding and using function composition to solve problems. The data was collected from three students in a precalculus course with a strong focus on developing students’ function understanding. The study consisted of individual interviews and a teaching experiment, with the goal of identifying the cognitive actions and understandings involved in responding to tasks that require students to string two function processes together for the purpose of relating two varying quantities. This presentation will report initial results from this study, and suggest implications for teaching and directions for further research.
(Self) Assessment: Aiding Awareness of Achievement
David E. Brown Utah State University david.e.brown@usu.edu 
In a sophomorelevel Linear Algebra class we concocted a scheme which we hope will inspire in students a regular reflection on content, and develop the habit of constructing their own examples. The scheme uses a selfassessment which is part of an advanced organizer, or agenda, that accompanies every lecture. Students are asked to complete the selfassessment as a high priority homework assignment and return it at the beginning of the following lecture. The selfassessment is constructed so that students respond via numeric responses intended to reflect their condencelevel for creating a certain example which corresponds to a content item from the lecture.
A Model Analysis of Proof Schemes
Model analysis is a method of quantitative analysis designed to examine students’ mental models of a concept. Furthermore, model analysis allows for the fact that students to be inconsistent with the models they use. Model analysis has been used with success in physics education, and it has great potential to be a valuable tool in exploring students’ concepts of mathematical proof. This presentation is a preliminary report on the use of model analysis to examine changes in students’ proof schemes. We will discuss how model analysis was adapted for use in examining students’ proof schemes, and preliminary results of a study on the changes in students proof schemes over the course of an introductory proof writing class.
Teaching Dynamic Optimization, a Research Based Proposal and its Testing

This report is concerned with the analysis of the first results obtained in a research project which consists in the design and analysis of results of an instructional approach to teach Dynamic Optimization based on APOS theory. A genetic decomposition of the concepts involved in the course and their relationships was developed. Activities and tests were designed according to the genetic decomposition for the teacher to use in the course. Results from the analysis of students’ work on activities, tests and final examination are discussed. Results show that the concepts involved in such a course are difficult for students, but that the experimental course can be considered successful. These first results, however, need to be complemented with analysis of other instruments to obtain enough information to refine the genetic decomposition and the activities for a refinement of the genetic decomposition and a new implementation of the course.
SelfEfficacy, Calibration, and Exam Performance in College Algebra and Calculus I
Joe Champion
University of Northern Colorado

Can college students’ confidence in completing exam items be used to predict the students’subsequent exam performance? I describe two preliminary quantitative studies of the relationship between college students’ mathematics selfefficacy—confidence to perform a specific task under specific circumstances—and exam performance in the contexts of College Algebra (n = 128) and Calculus I (n = 119) at a midsized doctoral granting university in the Mountain West. Using multiple linear regression analyses, findings support students’ calibration, or accuracy of selfefficacy judgments, along with selfefficacy, as important predictors of exam performance. Implications for teaching introductory college mathematics include the potential value of providing multiple sources of performance feedback to struggling students.
Connecting Beliefs and Missed Opportunities: A Model for Graduate Student Instructors' Reflection on Teaching
There is an expressed need for structured reflection on Graduate Student Instructor (GSI) teaching practice (Austin, 2002). Aligned with Schoenfeld's theory of teaching (1998, 1999) and a framework suggested in Arcavi and Schoenfeld (2008), this study provides a model for such reflection in which we identify GSI beliefs from interviews, observe GSI/student interactions in video data, and draw connections between the professed beliefs and decisionmaking in interactions with students. Using a series of four examples, we illustrate how these beliefs shaped the interactions in such a way that leads to 'missed opportunities' – instances in which opportunities for students to make sense of the mathematics are not seized. Highlighting the connection between beliefs and missed opportunities is meant to be used as space for reflection to improve teaching practices.
An Example of a Nontraditional Pedagogy in an Abstract Algebra Class: Was it Reform Teaching
Tim FukawaConnelly University of New Hampshire tim.fc@unh.edu 
The present study describes a nontraditional abstract algebra course, with a particular focus on proofvalidation discussions. This study will seek to illustrate that the instructor made purposeful decisions that required students to take the primary responsibility for presenting and critiquing proofs the instructor's classroom actions will be examined to better understand what is necessary to create where students to have significantly more mathematical authority.
Moreover the study will explore the relationship between groupvalidation process and the individual proofvalidation practice. The study has instructional implications for how to teach proofvalidation that may add to our understanding of how to help students develope that skill.
The Relationship Between Missing Graded Course Work and Student Outcomes in Undergraduate Math Courses
Lorraine Dame
University of Victoria

What significantly influences missing graded homework and what relationship does missing graded course work have with failure outcomes? Data from an anonymous inclass survey was used to analyze the impact of math help and other factors at the University of Victoria (UVic) on the proportion of missing graded homework (student failed to submit the work). Course grade sheets for all students in these courses in the spring of 2007 were used to analyze the influence of the proportion of missing graded course work on failure outcomes. It seems reasonable to infer that an improvment in the factors that are predictors of a low proportion of missing homework would have a strong positive impact on student success rates if students who miss more course work are also signicantly more likely to fail.
Interactive teaching and computational mathematics: Promoting mathematical conceptualization and competency
Gary Davis University of Massachusetts Dartmouth gdavis@umasd.edu 
Sigal Gottlieb University of Massachusetts Dartmouth sigalgottlieb@yahoo.com 
We report on an innovative methodology and analytical approach to enhancing student conceptualization and mathematical competency. We do this by studying the effects of interactive teaching of computationally oriented undergraduate mathematics. Previous studies indicate that a more interactive approach to teaching has a number of positive effects on student motivation, engagement and learning. Emphasizing computational aspects of mathematics in undergraduate mathematics courses provides students with the opportunity to experiment, the need to analyze and compare algorithms, and data to reflect upon. Webbased blogging software allows students to develop their mathematical writing to be more in line with professional scientific standards, and encourages them to write coherent mathematical stories. The significance of focusing on the conjunction of interactive teaching and computational mathematics is that it is vital that mathematics majors are motivated to learn and understand at a deep level the new mathematics required of them as successful twentyfirst century professional mathematicians, scientists, and engineers. We describe a number of research questions and data we are currently collecting that addresses these questions.
Learning Proof by Mathematical Induction
Mark Davis
University of Northern Colorado

Richard Grassl
University of Northern Colorado


This qualitative study of six preservice secondary teachers’ perceptions and performance around mathematical induction indicates strengths and challenges for collegiate teaching and learning. We report on constant comparative analysis of student mathematical work and on two focus group interviews of 3 students each.
A Two Semester Observational Study of Teaching Practices and Interactions in MultiSection Undergraduate Mathematics Courses
Jessica Deshler
West Virginia University

This qualitative and quantitative study of multisection undergraduate mathematics courses at a research university employed systematic classroom observation over a period of two semesters to determine teaching practices that predict student success. Classes were categorized qualitatively after each semester based on the overall amount of interactions observed during class. Detailed data was also gathered during the study by the researcher and by course coordinators, also by systematic observation using a trial observation protocol. Instructor and student behaviors were monitored, as well as different instructorstudent interactions. Logistic regression analysis was used to measure the influence of student GPA, instructor, observed teaching practices and interactions on student performance and the significance of the researcher’s qualitative classifications. Classes categorized as ‘highly’ or ‘minimally’ interactive were statistically significant predictors of student performance for at least one semester, as were studentinitiated logistic interactions and both instructor and studentinitiated interactions requiring an academic explanation.
Epistemography: How to Know What Students Know, and are Supposed to Know
We need to know better what students know and what they are supposed to know. But it is less easy to do than it appears at a first glance, particularly when students shift from secondary studies to undergraduate mathematical studies. This question is at the core of a new theory called epistemography. Following epistemography, mathematical knowledge is categorized in conceptual, semiolinguistic, instrumental, about the rules of the game and identification. Epistemography allows us to accurately analyse what the students know or are supposed to know or to learn during their mathematical studies. An empirical study of first year university students’ knowledge in calculus is done, in order to better understand the phenomenon of massive students failure in undergraduate mathematics. We have got already some preliminary results, on the importance of oral statements about the rules on the mathematical game, and on the difficulties related to the calculus symbolic language.
Aspects of Proving in Group Problemsolving in an Undergraduate Calculus Classroom Context
Kellyn Farlow
University of Maryland

We report on an analysis of the group work of three undergraduate calculus students who worked together to make sense of a given proof problem, construct an appropriate proof, and then account for their progress to their TA. Three categories, synthesized from the existing literature focused on proof production, were used to analyze these students’ work: (1) students’ approaches to the production of proof; (2) the expectations that frame students’ production of proof, arising from their perceptions of context and ‘audience’; and (3) the status of different approaches or forms of proof in terms of intuitive understanding, determining truth and establishing validity. This analysis contributes to the existing literature by illuminating how aspects of students’ understandings related to proof and proving dynamically shape the construction of proof in authentic classroom practice, in particular, result in tensions that students encounter during proof production in a real classroom setting.
How does the Implementation of an Inquiry Oriented Curriculum for Instruction on Euler’s Method
Affect the Relationship between Intended Curriculum and Learned Curriculum?
In this research an analytic framework was used to examine the implementation of an Inquiry Oriented Curriculum for Instruction on Euler’s Method. We used video analysis, homework problems and student interviews to identify the relationship between intended curriculum and learned curriculum. The students learned differential equations using a set of Inquiry Oriented instructional tasks based as the intended curriculum (Rasmussen and Kwon, 2007). We provide evidence to support the hypothesis that a collegiate level instructor may implement the curriculum differently from the intended curriculum. Preliminary findings indicate that different implementation does not greatly affect the learned curriculum. Implications of this study suggest that instructors can add their personal touch and experiences to the curriculum and still accomplish the learned curriculum goals.
The Great Gorilla Jump: A Riemann Sum Investigation
Nicole Engelke California State University, Fullerton nengelke@fullerton.edu 
Vicki Sealey West Virginia University sealey@math.wvu.edu 
The Great Gorilla Jump is an activity that was designed to introduce students to the structure of the Riemann sum within the context of distance, velocity, and time. This context was chosen because of the familiarity the students were expected to have with these concepts. Students in a calculus class were given the velocity of a falling object (a gorilla wearing a parachute) at a discrete number of points (times) and were asked to approximate the distance that the gorilla fell. Students were videotaped throughout the teaching experiment, and data was analyzed using principles of grounded theory and a Riemann integral framework. Results showed that students had little difficulty with the concept of limit, but significant difficulty with the product in the Riemann sum. These unexpected results prompted changes to the activity and to the hypothetical learning trajectory.
Exploring the Infinite Process Inherent in Multivariable Limits
There has been significant study into the question of how students transform mathematical processes into coherent objects. The limit concept plays a special role in these studies since it is among the first mathematical processes students encounter which is infinite in nature. In this presentation I will report on a study using taskbased interviews to describe how students conceptualize multivariable limits. The results suggest that some students struggle to understand multivariable limits due to weaknesses in conceptualizing infinite processes. We will look closely at the process created by students to understand multivariable limits and how the infinite nature of this process created an obstacle to further understanding. We will also see how overcoming this obstacle created a fundamental change in the way several students view the concept of limit.
Student Mathematical Discourse and Team Teaching
Julie Fredericks Portland State University jfreder@pdx.edu 
Martha VanCleave Linfield College mvcleave@linfield.edu 
It has been proposed that student achievement in mathematics can be significantly improved by increasing the quality and quantity of meaningful mathematics discourse in the classroom. This research project explores student mathematical discourse in both undergraduate mathematics classrooms and summer institute classes for inservice teachers. Utilizing a discourse observation protocol, comparisons between the quality of discourse in these two settings led to a further investigation of the impact of team teaching on student mathematical discourse. The actions of the second instructor in the team teaching setting classified as pressing for clarification, extension, sidetrip, and highlighting, were found to enhance the level of student mathematical discourse.
Factors that Affect College Students’ Attitudes toward Mathematics
Many students have poor attitudes toward mathematics. This mixed methods study investigates factors that affect college students’ attitudes toward mathematics as well as what may be done to reverse or prevent poor student attitudes in the future. Ninetynine college algebra students completed a retrospective quantitative survey in order to amass numerical data and guide interview choices. Twentythree of the ninetynine students were interviewed to gain indepth knowledge of what factors affect their attitude as well as suggestions on improving these attitudes.
From this study, student attitudes are most affected by four external factors: the teacher, teaching style, classroom environment, and assessments and achievement. Additionally, one internal factor, individual perceptions and characteristics, also affect student attitudes. It is suggested that educators can affect the four external factors in order to influence the internal factor and, in turn, student attitudes.
Women’s Mathematics Experiences that Influence the Pursuit of Undergraduate Mathematics Degrees
This contributed research report outlines findings of a recent study (2008) conducted at a Canadian university that investigated the mathematics experiences of women currently enrolled in upper years of undergraduate mathematics degree programs. Through semistructured individual interviews, the study explored the women’s high school and university mathematics experiences. Specifically, the women were asked about the supports and challenges in their mathematics experiences related to the four dimensions of the study: family, peers, personal characteristics, and the formal education system. Findings highlight the women’s preference for applied mathematics, their value of caring relationships with mathematics professors, and various issues regarding feeling ‘othered’.
Students’ Perceptions of Institutional Practices: The Case of Limits of Functions in College Level Calculus Courses
This paper presents a study of the interactions between instructors’ and students’ perceptions of the knowledge to be learned about limits of functions in a college level Calculus course. I modeled these perceptions using a theoretical framework which combines elements of the Anthropological Theory of Didactics, developed in mathematics education, with a framework for the study of institutions developed in political science. While a model of the instructors’ perceptions could be formulated strictly in terms of mathematical praxeologies, a model of the students’ perceptions had to include an eclectic mixture of mathematical, social, cognitive and didactic norms. In the paper, I will describe the models and illustrate them with examples from the empirical data on which they have been built (final examinations from the past six years, used in the studied College institution, and specially designed interviews with 28 students).

Bernadette MendozaSpencer

We report on analytic inductive analysis of semistructured interviews with 10 women in doctoral programs in mathematics departments, 5 in collegiate mathematics education and 5 in mathematics. We are developing crosscase narratives that capture and communicate graduate student experiences around learning to teach college mathematics. The focus of the study is the nature of the development of ownership of teaching as a part of the experience in graduate mathematics programs.
The Structure of College Geometry Students’ Arguments in the Presence of Technology
Prior research on students’ uses of technology in the context of Euclidean geometry has suggested technology can be used to support students’ development of formal justifications and proofs. This study examined the structure of students’ arguments while they used a dynamic geometry tool, NonEuclid, as they solved problems involving objects and relationships in hyperbolic geometry. Eight students enrolled in a college geometry course participated in a taskbased interview about properties of quadrilaterals in the Poincaré disk model. Toulmin’s argumentation model was used to analyze students’ uses of technology in the process of constructing arguments.
Calculus Students’ Routines of Function Composition
Aladar K. Horvath
Michigan State University

This paper focuses on calculus students’ routines of function composition within the context of algebraic, tabular, and graphical chain rule problems. Previous research on the chain rule has indicated that function composition is an important concept for understanding the chain rule. Because of the significance of function composition to the chain rule, these kinds of tasks provide a useful context in which to study function composition. Sfard’s (2008) commognitive framework, which combines communication with cognition, was used to identify these routines. The results revealed that when solving these tasks one third of the participants used function composition, another third used function multiplication instead of function composition, and the remaining third used function composition for algebraic tasks and function multiplication for tabular and graphical tasks. Additionally, students who had used function composition frequently verbalized “times” when explaining their solutions when they were performing function composition and not multiplication.
Preservice Secondary Mathematics Teachers’ Beliefs and Content Knowledge of Probability
The increased importance that has been placed on the inclusion of probability in the K12 curriculum has led to a need within undergraduate mathematics education to address how to teach probability. The study of preservice teachers (PST) beliefs and content knowledge of probability is a necessary first step towards addressing this need. This preliminary research report discusses results of a qualitative study involving pedagogically oriented task based interviews with 5 preservice secondary mathematics teachers.
Preparing to Teach Mathematics with Technology: Lesson Planning Decisions for Implementing New Curriculum
Sarah Ives
SarahIves@gmail.comNorth Carolina State University 
Hollylynne S. Lee North Carolina State University Hollylynne@ncsu.edu 
Teachers make decisions about how best to teach new curricula. The way the curriculum is written, the way teachers plan to teach it, and how it is actually implemented varies. This preliminary research report addresses the decision making process in planning to teach a probability and statistics unit within a technology methods course. Weekly planning and debriefing meetings between the instructor and the teaching assistant (who is the third author) were attended by the lead author and audio taped. Decisions included how to structure the instruction – teacher directed or in small groups; what questions to address in small groups or as a whole class; and what sections to include in teaching and which sections to assign as homework.
Psychometric Models and Assessments of Teacher Knowledge
Andrew Izsák
San Diego State University

Chandra H. Orrill
University of Georgia

Allan S. Cohen
University of Georgia

Jonathan Templin
University of Georgia

Joanne Lobato
San Diego State University

The proposal examines three different ways to combine descriptions of mathematical knowledge and psychometric models when building assessments that can be used at scale and that are informed by results from casestudy research on mathematical thinking. The three examples are sequenced so as to trace the increasingly finegrain size at which researchers have recently tried to measure teachers’ mathematical knowledge using psychometric models. The examples concentrate on inservice teachers but illustrate approaches that could be used to developed assessments for undergraduate mathematics courses.
Students’ Intuitions and Informal Understandings of Margin of Error and Confidence
Estrella Johnson
Portland State University

Sonya Redmond
Portland State University

Joanna Bartlo
Portland State University

Jennifer Noll
Portland State University

We report on a study in which we explored tertiary students’ informal and intuitive understandings of margin of error and confidence. This preliminary study reveals different ways in which these students thought about margin of error and confidence. Some of these ways may interfere with making sense of these ideas in their intended mathematical way, whereas some could potentially be leveraged to help students develop ideas that are consistent with thestatistical meanings of the terms. In the presentation we will discuss students’ ways of thinking about margin of error and confidence that emerged during our study, as well as questions about these ways of thinking that emerged for us as a result of our preliminary analysis. We will also discuss implications for teaching and questions for future research that arose as a result of this analysis.
Impact of Professional Development on the Classroom Practices of Adjunct Faculty
Matthew G. Jones California State University, Dominguez Hills mjones@csudh.edu 
Gwen Y. Brockman California State University, Dominguez Hills gbrockman@csudh.edu 
This project examines the impact of a professional development workshop on five of its volunteer adjunct instructorparticipants. Participants were observed teaching their courses during the following year, and surveys were collected from the instructors and their students. Preliminary findings reported based on the first semester of observations revealed that all instructors were implementing or attempting to implement cooperative learning, while just 3 of 5 were emphasizing multiple representations and using highlevel questions with their students. The authors will present qualitative and quantitative findings based on triangulation of observation data with instructor selfreports and student surveys, including data trends over time.
Function, Visualization, and Mathematical Thinking among College Students with Attention Deficit Hyperactivity Disorder
April Judd
University of Northern Colorado

For many people, AD[H]D is not a disorder but a trait, a way of being in the world. When it impairs their lives, then it becomes a disorder. But once they learn to manage its disorderly aspects, they can take full advantage of the many talents and gifts embedded in this sparkling kind of mind. (Halloway & Ratey, 2005, p. 4)
Lesson Study in Undergraduate Calculus: What Can We Learn About Teachers and Teaching from Lesson Study?


This is the first of two linked papers about one Lesson Study project in undergraduate calculus. In this study, mathematics education graduate students enrolled in a course called “Teaching College Mathematics” enacted a Lesson Study in a first semester undergraduate calculus class. The topic of the lesson was the use of different representations and methods to optimize functions. This paper will begin with a brief overview of the lesson study process in general and a description of how lesson study was enacted by the research group. The remainder of the paper is devoted to discussing the experiences of the instructor described from two perspectives. First, the lesson instructor discusses his attempts to encourage and monitor student engagement and motivation in the lesson. The session concludes with a summary of effects of the lesson study process on a graduate student TA’s thinking about teaching.
An Investigation of One Instructor’s Mathematical Knowledge for Teaching: Developing a Preliminary Framework
Karen Keene
North Carolina State University

Todd Lee
Elon University

Hollylynne lee
North Carolina State University

Megan Early
North Carolina State University

Peter Eley
WinstonSalem State University

Krista Holstein
North Carolina State University

Recent literature (i.e. Shulman, 1986; Hill, Schilling, & Ball, 2004) proposes a characterization of the kinds of mathematical knowledge used in teaching, but to date has provided limited specific information of what contributes to teaching inquiry oriented mathematics at the university level. Arguably, an essential component of enacting this type of course effectively is the instructor’s content knowledge and how it is incorporated into classroom discussions. We conducted a classroom teaching experiment with one instructor of an inquiryoriented differential equations course to explore what mathematical knowledge for teaching (specialized content knowledge, common content knowledge, and knowledge of mathematical horizon (Hill, Ball and Schilling (2008))) college instructors draw upon while facilitating classroom discussions. Eventually, we will develop a framework for mathematical knowledge used in teaching at the university level that will allow for future investigation with other mathematics instructors.
PreCalculus Students’ Interactions with Mathematical Symbols
This study presents information regarding the relationship between students’ understanding of mathematical symbols and their problem solving strategies. Confusion around certain problematic symbols and anticipation for certain symbols to be contained in the result can influence precalculus students’ problem solving goals and activities. Students create their own techniques for solving and producing expected results. Teachers may not always be aware of different goals with which students approach problem solving situations.
Appropriating New Definitions: The Case of Lipschitz Functions
Learning new definitions is a vital skill for mathematics students. This paper seeks to describe several ways that junior mathematics students approached a new definition in a real analysis context. Students explored the definition by sketching graphs, rewording the statement, looking for examples, negating the statement, finding nonexamples and relating it to definitions students had already encountered. Some of these tasks proved more fruitful than others.
Teaching Math Majors How to Teach
Yvonne Lai
University of Michigan, Ann Arbor

Marion Moore
University of California, Davis

Hillel Raz
University of California, Davis

We report on preliminary work with undergraduate math majors at a researchone university; the undergraduate students helped to teach an outreach and enrichment program for high school students. We were interested in how first time teaching affected the beliefs, attitudes, and knowledge of those with with significant mathematical background but little or no pedagogical training. During the pilot year, we found that the undergraduate participants were uniformly surprised that teaching requires mathematical and pedagogical foresight. Additionally, while they were curious about techniques such as group work, in practice, the participants were hesitant to use them out of lack of confidence. Our data, obtained through qualitative methods, suggest some potential priorities for initial work with undergraduates as teachers, and has implications on sequencing TA training programs to support a productive trajectory for novice graduate student instructors.
Recruiting, Supporting, and Graduating Women Mathematics Doctorates: Investigating the Community Elements of Successful Programs
Amanda Lambertus
North Carolina State University

This study is designed to help identify practices and strategies that encourage women to attend and graduate from university mathematics departments. It consists of three phases, the first two look at 15 different mathematics departments, providing a brief look at their recruitment policies and practices. The third phase examines three mathematics departments in depth. The researcher will spend two to three weeks at these schools. Wenger's (1998) communities of practice is the overarching framework of the study with pieces of the Carnegie Initiative of the Doctorate (Golde & Walker, 2006) adding a finer lens to the data analysis. The results of this collective case study will be reported both as the aggregate of the original 15 mathematics departments, and as a description of the three departments examined in depth.
HighAchieving Young Women’s Perceptions of Mathematics in College


Susan Bracken
North Carolina State University

The high achieving young women in this investigation are not choosing to study mathematics at the college level, despite being in advanced and college level mathematics courses throughout their high school careers. This study looks at their perceptions of mathematics and role it plays in their educational and career decision making processes. The interviews were coded using six different themes of mathematics as a tool. Each of the themes are supported from quotes from the participants. The results of this study may provide additional insights as to why young women are not pursuing advanced mathematics at graduate levels.
How Graphing Calculators and Visual Imagery Contribute to College
Algebra Students’ Understanding the Concept of Function
Rebekah Lane 
The purpose of this study was to answer the following research questions:
• What is the role of graphing calculators in understanding functions?
• How does visual imagery contribute to visual and nonvisual College Algebra students’ understanding of functions?
Interviews and document reviews were the data sets used in this study. The data were analyzed by using two theoretical frameworks: O’Callaghan’s (1998) translating component for understanding functions and Ruthven’s (1990) role of graphing calculator approaches. The investigation utilized the qualitative case study method.
A Local Instructional Theory for the Guided Reinvention of the Quotient Group Concept.
Estrella Johnson
Portland State University

In this paper, we describe a local instructional theory that has resulted from a series of design experiments focused on the quotient group concept. This local instructional theory will consist of 1) a generalized instructional sequence intended to support the guided reinvention of the quotient group concept and 2) a theoretical and empirical rationale for this generalized instructional sequence. We will describe the design experiments that informed the development of the instructional sequence and, in order to illustrate and motivate the local instructional theory, we will describe key aspects of the reinvention process in terms of the participating students’ mathematical activity.
Student Interpretations of the Equals Sign in Matrix Equations: The case of Ax=2x
During Fall 2007, we conducted a 4week classroom teaching experiment centered around the eigen theory unit in an inquiryoriented introductory linear algebra class at a public university in the southwestern U.S. About halfway through the course but prior to this unit, semistructured clinical interviews were conducted with eight of the 22 students in the class. One purpose of the interviews was to gain insight into the ways in which these students were thinking about ideas related to eigen theory. In this talk, we address the question: How do students think about the equals sign in the matrix equation Ax = 2x? We will discuss students’ conceptions of Ax, of 2x, and the ways in which different coordinations of these expressions reflect student conceptions of the equals sign in this context. Our talk will include case descriptions of three students’ conceptualizations that provide a representative sample of the variety of student responses.
An Analysis of College Mathematics Departments’ Credit Granting Policies for Students with High School Calculus Experience
Theresa Laurent 
This research investigated two questions related to mathematics departments’ credit granting policies for Calculus I based on students’ high school experience. First, the study determined common practice for departments to grant credit for Calculus I. Second, the research investigated if there are differences in students’ calculus achievement depending on the method by which credit was earned. To determine common practice, 244 mathematics departments were asked to complete a survey about their credit granting policies. A calculusbased placement test was administered to 143 college freshmen with high school calculus experience to determine their calculus achievement as they entered college. Using common credit granting methods, students were grouped based on the method by which credit was earned. Results indicated that students who earned credit by AP exams had significantly higher calculus achievement than their dual enrollment counterparts. In fact, dual enrollment students’ achievement was equivalent to students who did not earn credit.
Pedagogical Content Moves in an InquiryOriented Differential Equations Class: Purposeful Decisions to Further Mathematical Discourse


Krista Holstein
North Carolina State University

In this report, we are analyzing the ways in which a mathematics professor purposely promotes or furthers a mathematical discussion in an inquiryoriented classroom. In considering the relationship between mathematical content knowledge and pedagogical content knowledge, we specifically wish to further describe how one must draw upon content knowledge in an advanced mathematics classroom to make pedagogical decisions when facilitating a whole class discussion.
An Investigation of Mathematics Graduate Teaching Assistants’ Conceptions and Knowledge of Algebra in Relation to Their Teaching Practices
In this presentation the researcher will report the conceptual framework, design, and preliminary results of an ongoing project that investigates mathematics graduate Teaching Assistants’ conceptions and knowledge of algebra and how these are related to their current practices in teaching a precalculus college algebra course at a large and public university. A preliminary analysis of the existing data shows that the TAs put varied weights on four basic notions and utilities of algebra: Skills, Applications, Ways of thinking, and Foundations for advanced studies. The TAs’ conceptions of major factors for successful algebra teaching and learning share one common emphasis: practice and strengthening basic skills. Cultural backgrounds and past experience in mathematics learning explain partially the difference in the TAs’ teaching styles, mainly, the degree to which the TAs wanted to create an interactive classroom environment.
Investigating Student Approaches to Counting Problems: An Exploration Using the Notion of ActorOriented Transfer
Elise Lockwood 
The purpose of this study is to contribute to the literature in an area that is currently deficient: research on the teaching and learning of combinatorics at the undergraduate level. Adopting the perspective of actororiented transfer (Lobato, 2003), this research aims to capture critical elements of student approaches to solving a range of counting problems. Because the ability to detect structural commonalities among counting problems is often vital to their solution, gaining insight into students’ perspectives of problems stands to inform, and potentially improve, theteaching of combinatorics.
How to act? A question about encapsulating infinity
Ami Mamolo 
This report is part of a broader study that investigates the specific features involved in accommodating the idea of actual infinity. It focuses on the conceptions of two participants – a mathematics university student and graduate – as manifested in their engagement with a wellknown paradox: the pingpong ball conundrum. The APOS Theory was used as a framework to interpret their efforts to resolve the paradox and one of its variants. These two cases suggest there is more to encapsulating infinity than just the ability to ‘act’ on a completed object – rather, it is the manner in which objects are acted upon that is also significant.
What makes a textbook proof a satisfactory explanation for the reader?
Tyler Marghetis 
This preliminary research report describes an ongoing project investigating the factors that influence a reader's acceptance of a proof as a satisfactory explanation. This study examines one potential factor: the metaphorical language of the proof. When reading a proof involving continuity, does a reader's metaphorical understanding of continuity influence their acceptance of the proof as an explanation? I interviewed students in an undergraduate Analysis course to determine their metaphorical understanding of continuity. Most students understood continuity dynamically, speaking of \jumps" and movement. I will interview the subjects again, and present them with three proofs of the same result. While the proofs are structurally equivalent, they dier in the metaphorical language in which they are couched. The collection of empirical data is ongoing.
Incorporating InquiryBased Class Sessions with Computer Assisted Instruction
John C. Mayer
University of Alabama at Birmingham

Rachel D. Cochran
University of Alabama at Birmingham

Laura R. Stansell
University of Alabama at Birmingham

Heather A. Land
University of Alabama at Birmingham

William O. Bond
University of Alabama at Birmingham

Jason S. Fulmore
University of Alabama at Birmingham

Joshua H. Argo
University of Alabama at Birmingham

What is the effect of incorporating inquirybased group work sessions in a Finite Mathematics course in which the primary pedagogy is computerassisted instruction? Our research at a major state university investigates in a randomized study the relative effect of combining computer assisted instruction with, respectively, inquirybased group work sessions, traditional summary lectures of material to be covered in the computerbased part, and the latter combined with regular inclass quizzing on lecture material. The hypothesis is that inquirybased group work sessions differentially benefit students in terms of selfefficacy, content knowledge, and communications. Measures are described to demonstrate the effect.
Mathematicians, Mathematics Educators and High School Mathematics Teachers
Interpretations and Judgments Regarding High School Calculus Students’ Problem Solving
Allison McCulloch 
Mura (1995) pointed out the importance of casting light on the continuity of the influences that prospective teachers are exposed to during their training, in both mathematics and mathematics education courses. Since both mathematicians and mathematics educators work closely with prospective teachers to develop the knowledge needed to teach mathematics (Ball & Bass, 2003), it makes sense to compare what they attend to when drawing upon that specialized knowledge. The purpose of this paper is to share preliminary findings from an exploratory study aimed to compare what mathematicians, mathematics educators, and high school mathematics teachers (n = 37) attend to when making interpretations and judgments about calculus students’ problem solving methods.
Promoting Success in Applied College Algebra by Using Worked Examples in Supplemental Sessions
At a research university near the east coast, researchers have restructured an Applied College Algebra course by formatting the course into two large lectures a week, an active recitation size laboratory class once a week, and an extra day devoted to Supplemental Practice (SP). SP was added as an extra day of class where the SP leader only works with students on problems that were covered in the previous week’s class material. The researcher has worked with students in three different ways based off of workedout example research: 1) Active problem session, 2) Question and answer session, and 3) Hybrid method. The results have shown that SP participants have performed significant better than nonparticipants. In addition, student’s success in the course increases as the number of days attended in SP increases with close to 90% of the students being successful when they attend 11 or more of the 14 days.
Participant’s Perceptions and Experiences with WorkedOut Examples in Calculus
At a research university in the southwest part of the United States, student’s voluntarily attended two hour twice a week sessions that focused on students actively working examples using workedout example research. Students began by reviewing a workedout example on a particular topic and then helped the discussion leader work another example by telling the leader how to work specific steps. Finally the participants would work another problem by themselves. The student’s perception and experiences of the workedout examples show that workedout examples played an integral part of their success in the course. Their description of how the workedout examples helped them will be discussed along with quantitative data that backs up these perceptions and experiences. Finally, student’s perceptions and experiences are discussed that show that student’s confidence and selfefficacy is increased throughout the study
The Role of Quantitative Reasoning in Solving Applied Precalculus Problems
Kevin Moore Arizona State University Kevin.Moore@asu.edu 
Marilyn Carlson Arizona State University Marilyn.Carlson@asu.edu 
Michael Oehrtman Arizona State University Michael.Oehrtman@asu.edu 
This paper presents results from studying the quantitative reasoning abilities of 6 precalculus students as they attempted to solve novel word problems. We define the notion of quantity as a conceived attribute of something (e.g., the length of a side of a box) that can be committed to a measurement process. The data revealed that successful students were more likely to be engaged in i) initially reading (and rereading) the problem (while making meaning of) quantities to be related; ii) identifying both the varying and fixed quantities in the problem context; iii) constructing a diagram for the purpose of building a mental model of the quantities and their relationships, and iv) constructing new quantitative relationships as needed to relate and covary the desired quantities. Students who did not make progress in developing a formula appeared to have difficulty imagining the variation of quantities and their relationships.
An Investigation into Precalculus Students’ Conceptions of Angle Measure
Kevin C. Moore 
The presentation will report results from an investigation of three precalculus students’ conception of angle measure. The research to be presented discusses the results from a teaching experiment and individual exploratory interviews focused on students’ conceptions of angle measure, radian as a unit of measurement, and the unit circle. The subjects of the study were enrolled in a precalculus course that focused on developing students’ quantitative and covariational reasoning ability as foundational for understanding ideas of variable, rate of change, and function. Curricular activities engaged students in making meaning of applied problems, including the ability to identify varying and fixed quantities in an applied context, and formalizing the quantitative relationships expressed in the problems’ context. The results revealed that quantitative and covariational reasoning are foundational for understanding ideas of angle measure and for covarying angle measure with the yvalue of a point on the unit circle.
Effects of Inservice and Preservice Teacher Collaboration on Preservice Teachers’ Undergraduate Education
As part of a Methods of Teaching Secondary Mathematics course, preservice teachers spend eight weeks meeting on a regular basis in teams with inservice teachers. They have collaboration meetings, share responsibilities for leading and observing lessons in local middle school and high school classes, and participate in a variety of reflection activities in which they focus on improving their own teaching practice. Our research is aimed at documenting the effects this collaboration has on the preservice teachers’ preparation for teaching in the public school system prior to their student teaching experience.
Meaningful Collaboration in Secondary Mathematics and Science Teacher Professional Learning Communities


We present results from a fiveyear study of secondary science and mathematics teacher professional development involving collaborative Professional Learning Communities (PLCs). Over 200 hours of classroom and PLC video were selected for indepth analysis using techniques of open, axial, and selective coding. This analysis resulted in the emergence of three central categories of Process Behaviors characterizing the degree to which teachers inquired into topics of teaching and learning in a scientific manner and three categories of Dispositional Behaviors characterizing their approach to discourse. A central construct of Decentering emerged for following facilitator moves to manage the discourse, and we identified four categories of Teacher Beliefs that influenced the ability or willingness to change classroom practices. We also discuss relationships among these categories that emerged from the research and implications for design and implementation of teacher professional development.
Lesson Study in Undergraduate Calculus: What Can We Learn about Mathematical and Classroom Discourse from Lesson Study?
This is the second of two linked papers about one Lesson Study project in undergraduate calculus. In this study, mathematics education graduate students enrolled in a course called “Teaching College Mathematics” enacted a Lesson Study in a first semester undergraduate calculus class. This paper begins with a description of whether and how the multiple representations of the derivative, which are the focus of the study lesson, manifested in student discourse. The second part of the paper explores how the students in the class used the word, derivative, both in general and in two relationships: (1) between a function and its derivative function and (2) between the derivative function and the derivative at a point with graphs or algebraic expressions of functions. The paper concludes with a discussion of assessment and measurement issues associated with the use of lesson study as a research vehicle.
Conceptual Changes in Mathematics Majors’ Understanding of Completed Infinite Iterative Processes
Iuliana Radu
Rutgers, the State University of New Jersey

Keith Weber
Rutgers, the State University of New Jersey

In this paper, we report preliminary findings from a design teaching experiment whose goal was to explore ways in which students can come to reason about infinite iterative processes in a normative manner. Our data suggests that the students’ initial reasoning on infinite iteration was strongly influenced by two factors: 1) the belief that “completing” an infinite process involves “reaching the limit”, and 2) the students’ inclination to focus on global properties of the states (sets) produced by the process after a finite number of steps and generalize them to the “final state”. More importantly, we found that the manner in which the students continuously refined their understanding of completed infinite processes while working through a complex collection of infinite iteration tasks can be adequately explained by Wagner’s (2006) knowledge transfer theory.
Different Ways of Assessing the Persuasiveness of Mathematical Arguments
Juan Pablo MejiaRamos Rutgers University jp.mejia.ramos@gmail.com 
Matthew Inglis Loughborough University m.j.inglis@lboro.ac.uk 
Several recent studies have suggested that there are two different ways in which a person can proceed when assessing the persuasiveness of a mathematical argument: by evaluating the extent to which it is personally convincing, or by evaluating the extent to which it is publicly acceptable. In this presentation we use Toulmin’s (1958) argumentation scheme to describe a more detailed classification of the different ways in which students may assess the persuasiveness of an argument. We suggest that there are (at least) five different ways in which such an evaluation may take place. This classification is illustrated with data from an interview study that tracked the development of students’ argument evaluation behavior across the course of an undergraduate mathematics degree.
How Do You Know Which Way the Arrows Go? The Emergence and Brokering of a Classroom Math Practice
Chris Rasmussen
San Diego State University

Michelle Zandieh
Arizona State University

Megan Wawro
San Diego State University

The purpose of this report is to analyze how a particularly rich and complex inscription known as a bifurcation diagram emerged in an inquiryoriented differential equations class and how the teacher and some students functioned as brokers in this emergence. Adapting the work of Lave and Wenger (1991), we detail how these brokers facilitated others to become more central participants in a classroom math practice that features the creation, use, and interpretation of bifurcation diagrams. The significance of this case study is two fold. First, we move beyond reports of student difficulties in creating and using inscriptions and instead highlight students’ success at reinventing a complex inscription. Second, our analysis of the teacher’s role as broker highlights generalizable teacher moves that can facilitate students’ mathematical progress and enculturation into the practice of mathematics.
The Teacher Internship Experiences of Prospective High School Mathematics Teachers
Kathryn Rhoads Rutgers, the State University of New Jersey kerhoads@eden.rutgers.edu 
We describe the results of interviews with nine prospective high school mathematics teachers who had just completed their teaching internships. The results of this study show how the teaching philosophies emphasized in the prospective teachers’ education courses are at variance with the teaching practices of their cooperating teachers, and that the prospective teachers usually adopted the cooperating teachers’ teaching practices. The feedback that the prospective teachers received from their cooperating teachers and university supervisors predominantly concerned classroom management issues but rarely related to the mathematics content being taught.
Graduate Teaching Assistants Questioning Techniques
Kitty Roach
University of Northern Colorado

Nissa Yestness
University of Northern Colorado

This small qualitative study examined three graduate teaching assistants (GTAs) and how they used questioning techniques and strategies in their teaching. Data considered are video of two class meetings for each GTA and videorecorded interviews with each GTA. In between the two recorded class meetings, each GTA viewed and discussed a video case of college mathematics instruction concerning the concept of slope that illustrated various questioning techniques. The goal of the study is to explore the potential shadows of the videocase professional development activity on GTA self perceptions and classroom practice in the context of questioning techniques.
The nature of visualization and its impacts on the teaching and learning of the notion of continuity of functions.
This study examined how the visualization would influence students’ reasoning process while they dealt with the notion of continuity as well as uniform continuity. This presentation will illustrate classroom episodes in which visualization, using physical devices called epsilonstrips and deltastrips, was centered in the teaching and learning of the notion of continuity as well as uniform continuity of functions in advanced calculus courses.
On Conceptualizing Probabilistic Experiments and Quantifying Expectation: Insights from Students’ Experiences in Designing Sampling Simulation
Luis Saldanha
Arizona State University

Pat Thompson
Arizona State University

The design and use of sampling simulations figures prominently in the history of probability, both informing the mathematization of chance and the development of probability theory (Gigerenzer et al., 1989; Hacking, 1975; Stigler, 1986). In the last century sampling simulations played an important role in the development of statistical theory and other scientific fields; simulations were designed and used by scientists such as William Gosset and John von Neumann to create sampling distributions of variables about which little was known, and then probabilistic predictions about those variables were made on the basis of those distributions (Gigerenzer et al., 1989).
In educational settings simulation has been advocated as a potentially useful pedagogical tool to help students develop meaning for the concept of probability and for making informal inferences (Jones, Langrall & Mooney, 2007; Konold, 2002; Shaughnessy, 2007, 1992; Stohl & Tarr, 2002). The National Council of Teachers of Mathematics (NCTM, 2000) recommends that students in grades 912 should “use simulations to explore the variability of sample statistics from a known population and to construct sampling distributions” (NCTM, 2000, p. 324). But what is entailed in understanding and using simulations? Further, what might students experience in their efforts to design and use simulations to make informal inferences and draw probabilistic conclusions about situations that involve construing a stochastic experiment? Our paper will address these questions by reporting on part of a classroom teaching experiment that engaged a group of high school students in designing simulations within Prob Sim (Konold & Miller, 1996)—a sampling and probability simulation microworld.
Folding Back and Connecting to Historical Mathematical Concepts
The purpose of this qualitative study is to exemplify the characteristics of the growth of undergraduates’ mathematical understanding when connecting previously introduced concepts to the historical development of those concepts. Utilizing a key feature of the PirieKieren theory (1994b), the process of folding back, students revisit mathematical concepts they have already been exposed to in other mathematics courses as undergraduate mathematics majors. Through the process of folding back to their own inner levels of knowledge, students are provided opportunities to deepen and extend their understanding of mathematical concepts.
Mathematical Investigations in InquiryBased Courses for PreService Teachers
Tommy Smith
University of Alabama at Birmingham

Donna Ware
University of Alabama at Birmingham

Rachel Cochran
University of Alabama at Birmingham

Melanie Shores
University of Alabama at Birmingham

This study describes the efforts of a mathematics partnership in promoting inquirybased mathematics instruction in university mathematics courses and the resulting impact on mathematical knowledge and attitudes toward mathematics. The subjects for the study are preservice elementary and middle grades teachers taking a series of inquirybased mathematics courses. A variety of measures are used in determining participants’ knowledge of mathematics including objective tests, performance assessments, and portfolios. Additional measures such as classroom observations, focus groups, and surveys by external evaluators are used to measure changes in students’ attitudes toward learning mathematics in such an environment. Implications for changes in other university mathematics courses will be discussed.
Examining Mathematical Knowledge for Teaching in Secondary and PostSecondary Contexts
Natasha Speer
University of Maine

Karen King
New York University

This theoretical presentation highlights areas we believe need attention as the construct Mathematical Knowledge for Teaching (MKT), including Common Content Knowledge (CCK) and Specialized Content Knowledge (SCK), is generalized to secondary and postsecondary contexts. These constructs were developed in the context of research on elementary school teachers’ knowledge. Elementary teachers, however, typically differ from teachers of higher grades in their content preparation. We present a set of theoretical questions that arose from our examination of definitions of CCK and SCK as we attempted to utilize those definitions to characterize the nature of MKT at secondary and undergraduate levels. We illustrate these issues with data from two postsecondary mathematics instructional settings.
Natasha Speer
University of Maine

T. J. Murphy
Northern Kentucky University

Riding the Double Ferris Wheel: Students’ Creation and Interpretation of Trigonometric Functions in Realistic Settings
George Sweeney
San Diego State University

Trigonometry is an underrepresented topic in the mathematics education literature. This is unfortunate as trigonometry plays a significant role in applied mathematics, complex analysis, and other areas of advanced mathematics. I feel that understanding trigonometry requires the ability to move easily between different inscriptions and deal with covarying quantities. The purpose of this study is to better understand how students conceptualize trigonometric ideas and utilize them in dealing with realistic problem situations. I will briefly discuss a teaching experiment that I conducted in which students were asked to functionalize the height of a rider versus time as she rides a double Ferris wheel. We will discuss how students dealt with the covarying quantities that are inherent in the problem and also how they utilized the applet and their constructed graphs to create the function.
Finding a Suitable Alternative to a Potential Infinity Perspective: A Watershed Moment in the Reinvention of the Formal Definition of Limit
Craig Swinyard University of Portland swinyard@up.edu 
The purpose of this research was to generate insights into how students may come to understand the formal definition of limit of a function at a point, and to move toward the elaboration of a cognitive model of what might be entailed in coming to understand this formal definition. Evidence from the study suggests that students’ ability to reinvent the formal definition of limit was positively influenced by recognizing the need for, and subsequently finding, a suitable alternative to a potential infinity perspective (Tirosh, 1991). This report describes how one pair of students recognized the limitations of a potential infinity perspective, and employed the notion of arbitrary closeness as a means of spontaneously encapsulating the infinite limiting process.
Development of an Instrument to Measure Mathematical Sophistication
Jennifer Earles Szydlik University of Wisconsin Oshkosh szydlik@uwosh.edu 
Eric Kuennen University of Wisconsin Oshkosh kuennen@uwosh.edu 
Carol E. Seaman University of North Carolina Greensboro ceseaman@uncg.edu 
In this paper we propose a paperandpencil instrument to measure the mathematical sophistication of prospective elementary teachers. We call an individual mathematically sophisticated if her mathematical values and ways of knowing are aligned with those of the mathematical community based on nine interwoven traits involving patterns, structures, conjectures, definitions, examples and models, relationships, arguments, language, and notation. In other words, having mathematical sophistication means possessing the avenues of knowing of the mathematical community that allow one to construct mathematics for oneself. We will describe the development of the Mathematical Sophistication Instrument (MSI) and present the results of an initial study of its reliability and validity. We hope the MSI provides the mathematics education community a tool for measuring an important facet of teacher knowledge, and teacher educators a means for assessing pedagogies designed to teach students to think mathematically.
Beyond Static Imagery: How Mathematicians Think About Concepts Dynamically
Shiva Gol Tabaghi Simon Fraser University sga31@sfu.ca 
Nathalie Sinclair Simon Fraser University nathsinc@sfu.ca 
Researchers have emphasized the role of visualization, and visual thinking, in mathematics, both for mathematicians and for learners, especially in the context of problem solving (see Presmeg, 1992). In this paper, we examine the role that motion and time play in mathematicians’ conceptions of mathematical ideas, focusing on undergraduate concepts such as those found in linear algebra. In order to expand the traditional focus on (and distinction between) visual and analytic thinking (see Zazkis, Dubinsky, and Dautermann, 1996), we employ gesture studies, which have arisen from the more recent theories of embodied cognition. Expanding on Núñez’s (2006) work, we show how mathematicians’ gestures express dynamic modes of thinking that have been hitherto underrepresented.
Instructor Responses to Prior Knowledge Errors Made by Calculus Students
Jana Talley 
This study investigates the responses that Calculus I instructors have when assessing prior knowledge errors. A two part qualitative study consisting of student exams and instructor interviews was employed. Instructors of a summer Calculus I course were interviewed and asked to elaborate on exam grading decisions. Analysis of these interviews were used to develop additional interview questions for Calculus I instructors of various research and teaching backgrounds.
Prospective Elementary Teachers’ Multiplication Schema for Fractions
jtsay@utpa.edu 
shandy.hauk@unco.edu 
This research is to explore twelve prospective K8 teachers’ mental structures in twofactor multiplication with fractions. The results of this study are analyzed and presented via PirieKieren’s paths, on which the interviewees’ unconventional and problematic understandings were identified. Most of the problematic components appeared from the interviewees were the properties the interviewees noticed: multiplier, confined understanding of fraction as partwhole relation, fraction as multiplier, and fractionasmultiplier acting on a whole and/or fraction
number.
Teaching Proof by Mathematical Induction
Shandy Hauk
University of Northern Colorado


Bernadette MendozaSpencer
University of Northern Colorado

This qualitative study examined two mathematicians’ approaches to teaching proof by mathematical induction (PMI) to undergraduate preservice secondary teachers. Data considered in the study included classroom video of four weeks of instruction for each professor, a 90 minute interview with each instructor, focus group interviews of three students from each professor’s class, and student solutions to common final exam PMI items. We report on the nature of the knowledge for teaching of PMI of the two instructors.
Women with Advanced Degrees in Mathematics in Doctoral Programs in Mathematics Education
Allison Toney
University of North Carolina Wilmington
allison.toney@gmail.com 
I report on the results of my dissertation work, a qualitative investigation on the nature of the graduate school related experiences of women in collegiate mathematics education doctoral programs. I interviewed 9 women at 3 universities. Each woman had an advanced degree in mathematics and chose to move into a collegiate mathematics education doctoral program housed in a mathematics department. I used narrative and autoethnographic approaches, and consequently I was a coparticipant. The focus of the twointerview protocol was exploring and extending the framework for doctoral mathematics student experience suggested by Herzig (2004a, 2004b). Results support the existing framework offered in the literature, as well as the emergence of 3 new categories: self as scholar, "my teaching," and future possible self.
APOS Framework and Geometric Related Rates in a first course in Calculus.
Mathew Tziritas 
This research explores the use of the APOS framework to construct a didactical tool for students learning geometric related rates. The goal is to determine whether a student’s ability to sketch the problems with appropriate variable placement, as well as their distinction between variables and constants, can be improved using the ACE teaching cycle. This study will also
examine whether this didactical lesson will improve their overall schema of geometric related rates problems.
Student Motivations for Mathematical Understanding in an InquiryBased Calculus Classroom
Janet G Walter Brigham Young University jwalter@mathed.byu.edu 
Student motivation has long been a concern of mathematics educators. We present Contextualized Motivation Theory (CMT) as a means for understanding the complexities of student motivations in an inquirybased university honors calculus class. This qualitative, grounded theory study is part of a longitudinal project in calculus learning and teaching. Here, we characterize motivation, defined as an individual’s desire to act in particular ways, through analysis of students’ extended, collaborative problem solving efforts. Students persisted beyond obtaining correct answers to build understandings of mathematical ideas. Analysis of extended student collaborations suggests a supporting “web” of motivations, existing simultaneously, from which a learner chooses to act upon at any given time. Students chose to act upon various intellectualmathematical motives and personalsocialemotional motives. CMT positions personal agency as central, characterizes the social nature of motivation, and encompasses conceptually driven conditions that foster student engagement in mathematics learning.
The Influence of Risk Taking on Student Creation of Mathematical Meaning: Contextual Risk Theory (CRT)
Janet G Walter Brigham Young University jwalter@mathed.byu.edu 
The primary concerns of mathematics educators are learning and teaching mathematics. Atkinson (1957) asked, “What implications and benefits might there be if learning were perceived as a risktaking event?” (p. 266). Atkinson’s guiding question articulates the underlying motivation of this study: to analyze the risks students take in the mathematics classroom and how risk influences student creation of meaning and development of understanding. We define risk in the mathematics classroom to be any observable act that entails uncertain outcome. Contextualized Risk Theory (CRT) is introduced to improve our understanding of the risks students take in learning mathematics in a studentcentered classroom where students exercise personal agency in mathematical problem solving.
Task Design: Towards Promoting a Geometric Conceptualization of Linear Transformation and Change of Basis
Megan Wawro
San Diego State University
meganski110@hotmail.com 
Reformoriented mathematics education calls for instruction that assists students in developing from their current ways of reasoning into more complex and formal mathematical reasoning (e.g., Gravemeijer, 2004). Such instruction centers on a host of theoretical and pragmatic concerns. Our research explores many of these concerns in a first course in undergraduate linear algebra. Our work draws on the theory of Realistic Mathematics education to design instructional sequences that build on student concepts and reasoning as the starting point from which more complex and formal reasoning develops. This emphasis on creating a classroom environment that fosters students’ ability to reinvent advanced mathematical concepts served as inspiration for the development of an instructional sequence with the particular goal of fostering a geometric conceptualization of linear transformation and change of basis in R2. The proposed presentation will focus on a particular task from the aforementioned instructional sequence, the Change of Basis Task, that was developed with the intent of leading to a student driven reinvention of the wellknown equation A = PDP1, for the case where A, P, P1and D are 2x2 matrices. During the presentation, we will explore the Change of Basis task together, discuss pilot data in the form of student responses to the task that were collected during paired interviews, and elaborate upon what affordances the inclusion of the Change of Basis Task may provide in an upcoming semesterlong classroom teaching experiment (Cobb, 2000).
BEST PAPER 2009
Mathematics Majors’ Evaluation of Mathematical Arguments and Their Conception of Proof
Thirty mathematics majors were observed as they read and evaluated ten mathematical arguments. The results of this study suggest that: (a) mathematics majors do not hold empirical proof schemes, as is widely believed, but many are convinced by perceptual arguments, (b) they will often accept an argument as “mostly correct” and as a valid proof even if they do not understand a pivotal claim within the argument, and (c) many lack particular proofreading skills needed to recognize the flaws in some arguments.
The majority of mathematics educators would agree to at least one of the following statements. (a) If an argument is completely convincing to an individual or a community, then it is a mathematical proof (to that individual or community). (b) If an argument is not completely convincing to an individual or a community, then it is not a proof (to that individual or community). Based on data from interviews with ten mathematicians, as well as other findings in the mathematics education literature, I will argue both claims are false. Further, I will contend that our community’s acceptance of these claims has led us to draw inappropriate inferences from existing data and offer dubious pedagogical suggestions.
How Students Use Their Textbooks: Reading Models and Model Readers
Aaron Weinberg
Ithaca College

While most math classes require students to have a textbook, many students don’t use their book in ways that help them understand the mathematics. The ideas of reading models and the model reader are complementary theoretical perspectives that we can use to describe students’ relationship with their textbook. The former describes ways students approach reading, while the latter describes the competencies and dispositions required to read a text. The goal of this paper is to outline these theories, describe how we can use them to make sense of the ways students use math textbooks, and explore how we can use them to create strategies for improving student learning.
Measuring InquiryOriented Teaching in the Context of TA Professional Development
Ian Whitacre San Diego State University ianwhitacre@yahoo.com 
Susan D. Nickerson San Diego State University snickers@sciences.sdsu.edu 
We report on results of an analysis of mathematics teaching assistants’ (TAs) discursive moves in parallel segments of a research lesson, as taught by each of the TAs. The analysis employed Rasmussen, Kwon, & Marrongelle’s (2008) framework for interpreting inquiryoriented teaching. The research lesson was conducted in six sections of a mathematics content course for preservice elementary teachers, intended to promote inquiry. The analysis reveals significant differences in the TAs’ discursive moves with regard to the potential for the lesson to promote inquiry and desirable norms. These results may be of interest to those designing professional development programs for TAs, those interested in conducting lesson study at the university level, and those concerned with analyzing inquiryoriented teaching.
The Instructor's Important Role in Supporting Mathematical Arguements in a K5 Mathematics Specialist Program
Joy Whitenack
Virginia Commonwealth University

Laurie Cavey
James Madison University

Aimee Ellington
Virginia Commonwealth University

In this presentation, we use examples from a one lesson taken from an algebra course for K5 mathematics specialists to illustrate the instructor’s role in supporting argumentation. Krummheuer’s (1995) theory of ethnography was a particularly useful methodological tool for tracing the argument that unfolded during the discussion. As part of this interpretive model, we assume that normative ways of making mathematical arguments are socially accomplished, although individuals contribute to and participate in these arguments in different ways. Preliminary findings suggest that as the instructor highlighted teachers’ explanations he and the teachers established mathematical arguments. In some cases the instructor provided warrants or backings that remained implicit or omitted (cf. Yackel, 2002). In other instances the instructor coordinated different explanations to substantiate or validate the arguments that emerged. By doing so, he made it possible for teachers to engage in making more formal arguments.