for the Eleventh Special Interest Group of the Mathematical Association of America
 on Research in Undergraduate Mathematics Education

Conference on Research in
Undergraduate Mathematics Education

DoubleTree Hotel, San Diego - Mission Valley, California
February 28 – March 2, 2008


When is an "Impoverished" Model for a Mathematical Concept Best?
The Case of Prime Factorization

Susan Addington
California State University - San Bernardino
Madeleine Jetter
California State University - San Bernardino

A “rich” model of a mathematical concept is one that represents many aspects of the concept. An “impoverished” model represents at most one aspect. Preservice elementary teachers find it difficult to answer questions that can be solved efficiently by applying prime factorizations. A card game provides a metaphor for prime factorization that a) matches the intuitions that professional mathematicians use and b) affords a socially mediated learning experience. However, the card game does not model any aspects of multiplication other than as a binary operation. Pilot tests show increased achievement on simple yet telling questions such as “Is 53x57 prime?”


Explaining Student Success in One PDP Calculus Section: A Progress Report

Aditya P. Adiredja
University of California Berkeley
Randi A. Engle
University of California Berkeley
Danielle Champney
University of California Berkeley
Amy Huang
University of California Davis

Mark Howison
University of California Berkeley
Niral Shah
University of California Berkeley
Pegah Ghaneian
University of California Berkeley


We report initial findings from an intensive study of one especially successful PDP calculus section in order to investigate four hypotheses offered by program designers explaining students’ success in them. First, we found evidence that students productively engaged with one another’s mathematical ideas to solve worksheet problems in their small groups. Using videos of the small groups in action, we are currently investigating how the group norms developed over time. Second, we found that the section supports the incorporation of personal identity in the development of their mathematical identity, with students increasingly identifying with mathematics over the course of the semester while continuing to report that they could "be themselves" in section. However, less evidence currently supports the hypotheses that students' learning was mediated by engagement in especially challenging problems or that students' self-efficacy beliefs were strengthened by successfully solving such problems.

Scaling Up Instructional Activities: Lessons Learned from a Collaboration between a
Mathematician and Mathematics Education Researcher

Joanna Bartlo
Portland State University
Sean Larsen
Portland State University
Elise Lockwood
Portland State University

We report on a study in which we explored how a mathematician made sense of a research based abstract algebra curriculum while implementing it for the first time. Our goal in this study is to investigate what types of information are useful to an instructor not familiar with this material. This preliminary study reveals some difficulties an instructor might have with this curriculum and what supports might be needed for it to be implemented with fidelity. In the presentation we will discuss the lessons we, as curriculum designers preparing to design instructor support materials and to begin to more broadly implement the curriculum, learned from working with and trying to support a mathematician in his attempt to make sense of and implement the curriculum. We will also discuss implications for the design of teacher materials that accompany this curriculum.


Discourse Analysis: The problematic analysis of unstructured/unfaciliatated group discussions

Jason K. Belnap
Brigham Young University
Michelle Giullian
Brigham Young University

An increasing number of researchers are studying discourse in order to understand classroom instruction and online discussions. Based on social linguistics and activity theory, researchers have derived frameworks for breaking down discussion and identifying its structure and composition. These frameworks can be fairly easily utilized to analyze both class instruction and asynchronous online discussions, because they are typically either highly facilitated or well structured.

In a recent qualitative study, we encountered significant challenges in applying these frameworks to discourse in a professional development (PD) setting with little facilitation. In this presentation, we explore the challenges faced when conducting discourse analysis in unstructured discussions by: first, reviewing existing analytical frameworks; second, identifying specific problems that we faced; and third, illustrating our attempts to overcome them in our own research study. We hope that this will open discussion regarding other ideas or ways for visualizing and analyzing discourse in similarly complex settings.


Teaching Discussion among Graduate Mathematics Teaching Assistants: Elements
contributing to teaching discourse

Jason K. Belnap
Brigham Young University
Michelle Giullian
Brigham Young University

Concern regarding the quality of undergraduate mathematics instruction has drawn attention to the professional development of graduate mathematics teaching assistants (GM-TAs), who often fill the role of instructor rather than assistant. Consequently, many programs have been developed to support GMTAs teaching development. Often these programs rely upon teaching dialogue among GMTAs; hence, the successful preparation of GMTAs in part depends upon their ability to carry-out productive discussion regarding teaching.

Using one discussion-based GMTA program, we studied GMTAs’ teaching discourse. Through this study, we identified various elements on which GMTAs relied to contribute to teaching discussions. These included past and current teaching experiences, discussions with peers and faculty, cognitive concepts developed through their apprenticeship of observation, and interests and skills that they personally bring with them. GMTAs utilized these elements differently throughout their discussions, such as for clarification, justification, and control. This paper/presentation details these elements, their roles, and resulting implications.


The Development of Covariational Thinking in a College Algebra Course

Stacey A. Bowling
Arizona State University
Kevin C. Moore
Arizona State University

This presentation describes the emerging understandings and covariational reasoning behaviors of eight students in a reformed college algebra course. This research was situated in the context of a larger project to redesign the curriculum and instruction for a large-enrollment college algebra course. The primary goal of this redesign effort was to build students’ understanding of, and ability to use, central concepts of precalculus by taking a covariational approach to teaching ideas of variable, rate of change, function, function composition, function inverse, and exponential growth. Student behaviors are analyzed using Carlson’s (Carlson et al, 2003) covariation framework. Initial results suggest that after participating in such a course, many students are able to exhibit behaviors indicative of an improved understanding of covarying quantities.


The Development and Evaluation of a Program for Improving and Assessing the Teaching of Mathematics and Statistics

David E. Brown
Utah State Univeristy
Brynja Kohler
Utah State University
James Cangelosi
Utah State University

We report on a program intended to improve and assess teaching practices in our mathematics and statistics department, and on results from the program’s initial field tests. The structure of our program was influenced by current legal standards for “evaluation of personnel” that have been established through a string of litigations occurring over the past 25 years. Our program works as follows: an instructor employs two disjoint teams, formative and summative, which provide their respective recommendations and evaluations under the protection of a data curtain (teams are kept ignorant of each other’s activities), and all operations and logistics (including maintenance of the data curtain) are overseen by a third team. What we find notable is the measurably positive effect the process has on all involved, and the program’s ability to accommodate a variety of teaching styles and objectives. Our evidence suggests our program is comprehensive and notably constructive for participants.


Exploring Epistemological Obstacles to the Development of Mathematic Induction

Stacy Brown
Pitzer College

Research on undergraduates’ understandings of proof by mathematical induction (PMI) has shown that undergraduates experience difficulty with this proof technique (e.g., Dubinsky, 1989; Movshovitz-Hadar, 1993). Harel and Sowder (1998) and others (Author, XX), however, have questioned the extent to which these difficulties are due to traditional instructional approaches that tend to hastily introduce the definition of mathematical induction and do not facilitate the development of PMI as a means to solve a class of problems. In an effort to distinguish between those difficulties that are didactical in nature (that is, due to instructional choices) and those that are epistemological (that is, whose origin is the concept itself), this paper will examine findings from two teaching experiments. The first involved undergraduate mathematics and science majors. The second is ongoing and involves advanced 6th grade students. The purpose of the paper is to explore similarities and differences in the students’ approaches to PMI-appropriate tasks and then to use the multi-age comparison to evaluate potential epistemological obstacles to PMI.


The Effect of Trigonometric Representations on Mathematical Transitions

Patricia Byers
York University

When students select courses in secondary school rarely do they realize the impact these choices have on post-secondary school studies. This research investigates student achievement in college technical mathematics based on the transition students make from secondary school mathematics studies and the impact teaching trigonometry has on student learning. The goals of the mixed-methodology study are two-fold: (1) to provide a deep examination of the data, establishing a statistical framework available for comparison research in future studies; and, (2) to develop a comprehensive analytical framework on which to examine mathematical representations, specifically trigonometric representations, in secondary school and college classrooms.


Graduate Teaching Assistant Instructor Expertise and Algebra Performance of College Students

Karla Childs
Pittsburg State University

This longitudinal study examined the relationship between level of GTA instructional expertise, amount of GTA teaching experience, and academic performance of their college algebra students measured by course grades.  College algebra grades for all students in classes taught by GTAs over six years and 43 sections were analyzed (n = 2198).    A chi-square analysis indicated there was a statistically significant relationship between Trained (Yes or No) and Years (1 or 2) on withdraws from college algebra.            

GTAs with two years of training and two years of experience had significantly fewer students withdraw from their courses than GTAs in their first year of teaching or GTAs that were not trained.  Results of the present study indicate that a well-planned program of support and professional development for graduate students in the role of teaching assistants combined with experience appears be a major factor in improving academic persistence for students in college algebra.


Documenting “Speaking with meaning” in a College Algebra Course

Phillip G. Clark
CRESMET/Scottsdale Community College
Kevin Moore
CREMSET/Arizona State University
Kate Mullin
CRESMET/Arizona State University

The purpose of this research is to describe the emergence of the sociomathematical norm of speaking with meaning (Carlson, Clark, & Moore, In Press) and delineate how a college algebra instructor helped enable this emergence. Speaking with meaning has the dual nature of being both a sociomathematical norm regarding what constitutes sufficient mathematical participation as well as being a tool that can be used in the classroom to elicit such participation.  Preliminary analysis shows that attention by the teacher to student responses is enabling them to speak more meaningfully. In the case of this college algebra course the students are able to explain functions in terms of inputs and outputs. Thus, in this class, to speak with meaning about functions means to couch responses about functions in terms of input and output.


What is Mathematics: Student and Faculty Views

Jacqueline Dewar
Loyola Marymount University

The questions this preliminary research explores are: (1) How undergraduate STEM students’ understandings of mathematics compare to an expert view of mathematics, and (2) whether a single course can enhance future teachers’ views of mathematics. Written responses to “What is mathematics?” from 55 STEM students, 7 future teachers, and 16 mathematics faculty revealed that most students see mathematics as being the study of a list of topics (primarily numbers) and applications. On the other hand, for faculty, mathematics encompasses pattern, proof, logic, abstraction, and generalization in addition to applications. Hardly any students (initially) considered mathematics to involve abstraction or generalization. The responses gathered from a small group of future teachers before and after a particular course along with additional evidence from the study indicate that a single course can nudge future teachers toward a more expert view of mathematics.


College Physics Majors’ Mathematical Thinking and Problem-Solving Skills

Barbara Edwards
Oregon State University

The importance of good mathematical problem solving skills is significant for learners in many settings – among them the physical sciences. This research investigates the problem-solving skills and mathematical thinking of advanced physics and physics engineering students and physics and mathematics faculty – categorizing their thinking as primarily geometric, analytic, numeric or harmonic (based roughly a framework developed by Krutetskii (1976). This talk presents an analysis of one interview task and the results of several interviews with students and faculty who engaged in this task, trying approaches – some successful and some unsuccessful.


College Students’ Understanding of Rational Exponents: A Teaching Experiment

Iwan R. Elstak
Georgia State University

College students understanding of rational and negative exponents is examined, followed by a teaching experiment to test an alternative trajectory for teaching rational exponents.  Rates of change, factors of multiplication and repeated multiplication are used as a basis. Roots are presented as ‘fractions’ of the base and ‘decimal’ roots are used to calculate decimal exponents. Numerical, graphical and diagrammatic tools illustrate the process. Interviews, worksheets, video and audio taping documented the students’ evolution.

Results suggest that the definition of exponents students learn in school provide the primary lens for conceptualizing rational and negative exponents. The laws of exponents play no foundational role in this process. Obstacles encountered were additive models of thinking about rates of change and slow understanding of factors of multiplication.

Post-interview questionnaires with the same content as pre-interview questionnaires showed improved responses on most questions.


Developing the Solution Process for Related Rates Problems Using Computer Simulations

Nicole Engelke
California State University - Fullerton

Related rates problems are a source of difficulty for many calculus students. There has been little research on the role of the mental model when solving these problems. Three first semester calculus students participated in a teaching experiment focused on solving related rates problems. The results of this teaching experiment were analyzed using a framework based on five phases: draw a diagram, construct a functional relationship, relate the rates, solve for the unknown rate, and check the answer for reasonability. A particularly interesting aspect of the relate the rates phase was the development of what the students called “delta equations.” The creation of the delta equation differs from a traditional approach to solving related rates problems and may facilitate the students’ understanding of the solution process.


In-Service Teachers’ Proof Schemes in Transition

Evan Fuller
University of California - San Diego
Osvaldo Soto
University of California - San Diego & San Diego State University
Guershon Harel
University of California - San Diego
Alfred Manaster
University of California - San Diego

The goal of this research is to examine cognitive, social, and instructional aspects in the transition between proof schemes: from the external conviction and empirical proof schemes to deductive proof schemes, focusing, in particular, on the transition from Result Pattern Generalization to Process Pattern Generalization.  Of particular focus is the instructor’s way of implementing DNR in his attempt to facilitate this transition. Preliminary findings indicate that Empirical proof schemes are resistant to change. However, there is evidence to believe that a focus on causality over a long period of time can prove effective. The teaching practices that have been used to instantiate this focus on causality are also reported here.


How Blending Illuminates Individual and Collective Understandings of Calculus

Hope Gerson
Brigham Young University
Janet Walter
Brigham Young University

Conceptual blending is gaining momentum amongst mathematics educators interested in better conceptualizing mathematical meanings students are building. We used conceptual blending as a lens to illuminate individual and collective understandings of calculus concepts as they emerged during sustained mathematical inquiry. We share some of the insights we have gained by using this lens in our analysis. Viewing the mathematical connections along with the emergent structure that follows allowed us to more fully characterize students’ constructions of meaning for mathematics. Additionally we have found that conceptual blending is flexible in the unit of analysis (it can be used to analyze conversations among group members or single utterances), brings to the forefront elements of the input and blended spaces and the connections between them, emphasizes the meaning that students are building for important mathematics, and aids comparisons between conceptions held by a student or different students.


Implications of Undergraduates’ Conceptions of Function

Todd A. Grundmeier
Cal Poly - San Luis Obispo
Jacey Branchetti
Cal Poly - San Luis Obispo
Joaquin Castillo
Cal Poly - San Luis Obispo
Carla Scherer
Cal Poly - San Luis Obispo

This study explored university students’ conceptions of function by focusing on their abilities to define and apply the concept of function.  A survey was administered to 289 undergraduate students from varying grade level and major.  Results focus on the participants as divided into three groups: Pre (had not taken Methods of Proof), Current (taking Methods of Proof), and Post (taken Methods of Proof).   The survey results suggest that all three groups had difficulty defining function and a participant’s ability to define function was not a predictor for their ability to provide a real world example or recognize functions. Additionally the survey results suggest that there is a lack of retention of the concept of function as participants take upper division mathematics courses.


Prospective Secondary Mathematics Teachers’ Conceptions of Rational Numbers

Todd A. Grundmeier
Cal Poly - San Luis Obispo
Jenna Babcock
Cal Poly - San Luis Obispo
Sarah Odom
Cal Poly, San Luis Obispo

This research explored the rational number understanding of prospective secondary mathematics teachers.  The research aimed to determine if future teachers have misconceptions about rational numbers that are consistent with those shown by researchers to exist in students and teachers.  The exploratory study included a survey completed in an interview setting with four junior or senior mathematics majors who intended to become middle or high school teachers.  The results of the data analysis suggest that these prospective teachers struggled to correctly answer questions about rational numbers when they could not rely on procedural or past knowledge and had difficulty relating their definitions of rational and irrational numbers to solutions of problems.  Also, participants’ misconceptions were highlighted through their general difficulty understanding part-whole relationships, making comparisons between abstract rational numbers, visualizing problems and writing word problems.


Implications of history for mathematics education: The case of limit

Beste Gucler
Michigan State University

This presentation takes as a basis prior research on calculus teachers’ knowledge of student thinking in limit and builds on it by investigating the historical development of the concept and its implications for the teaching of limit. After presenting the work on the historical development of the concept, I will discuss how this historical analysis gives us significant information in terms of the prerequisites as well as the teaching of the concept. Finally, I will seek audience members’ suggestions for how to develop methods to gain better information about calculus teachers’ knowledge of their students’ thinking about limit in light of this new perspective.


Women with advanced degrees in mathematics in doctoral programs in mathematics education

Shandy Hauk
University of Northern Colorado
Alison Toney
University of Northern Colorado

We report on analytic inductive analysis of interviews with 8 women with advanced degrees in mathematics who chose to move into doctoral programs in mathematics education (specifically, doctoral programs in mathematics education housed in mathematics departments). The participants are in doctoral programs at 3 different universities. The focus of the two-interview protocol is exploring and extending the framework for doctoral mathematics student experience suggested by Herzig (2004a, 2004b). Preliminary coding of data indicates the emergence of several additional themes not previously suggested in the literature, as well as a need to refine the language for some of the existing themes.


Translating Information from Graphs into Graphs: Signals Processing

Margret A. Hjalmarson
George Mason University
John R. Buck
University of Massachusetts - Dartmouth
Kathleen E. Wage
George Mason University

Students studying signals and systems processing participated in clinical interviews related to their understanding of fundamental concepts in the discipline. This includes the interpretation of graphical representations of signals and functions. Preliminary analysis indicates that students with understanding of fundamental structures in signal processing (e.g., frequency, magnitude) can organize information from multiple graphs simultaneously to make projections about a system of signals. Mathematically, they need to be able to organize and interpret multiple representations in order to make predictions about a system.


Looking at calculus students’ understanding from the inside-out:
The relationship between the chain rule and function composition

Aladar Horvath
Michigan State University

The chain rule is an important topic of calculus that has received little attention in the mathematics education literature. This report includes results from an exploratory study where students enrolled in first semester calculus were given tasks involving the chain rule. The results revealed that these students replaced function composition with function multiplication for functions that they had experienced in precalculus, but had not yet encountered in calculus (e.g., exponential, logarithm, and inverse trigonometric).  The discussion portion will focus on studying students’ understanding of function composition through the lens of chain rule problems and the tasks designed to address this issue.


Computer Algebra Systems (CAS) in University Mathematics Instruction: A Preliminary
Research Report Investigating CAS Technology Usage and Sustainability

Daniel H. Jarvis
Nipissing University
Zsolt Lavicza
The University of Cambridge
Chantal Buteau
Brock University

The use of Computer Algebra Systems (CAS) is becoming increasingly important and widespread in mathematics research and teaching at the university level. Notwithstanding, there exists very little in the way of formalized support presently in place to assist: (i) university mathematics instructors who wish to move forward in the area of technology for teaching; and, (ii) university mathematics departments that wish to sustain the use of CAS-based software over time . Furthermore, in contrast to the large body of research focusing on technology usage which exists at the secondary school level, there is a definite lack of parallel research at the post-secondary level. In this paper, we will report on an ongoing international research project focusing on technology usage in undergraduate mathematics instruction. Three researchers from Canada and England are in the process of conducting a national survey of technology usage, and collecting data from several case study sites wherein university mathematics departments have successfully incorporated technology into their respective mathematics programs over time. Our research framework and progress will be shared and further input/ideas from international colleagues will be sought out during this presentation.


Opportunities to learn mathematics for teaching at community colleges

Amy Jeppsen
University of Michigan

This is a preliminary report on a study to evaluate opportunities for elementary education students to learn mathematics at the community college level. Recently, there has been a great deal of interest in the mathematical preparation of prospective teachers. One site in which this learning is intended to take place is the mathematics course or sequence of courses required for certification, and yet very little research has addressed the opportunities available for students to learn mathematics in this setting. Still less is known about the equivalent course at the community college level, where a large proportion of students fulfill their mathematics requirements. This study uses the textbook as a site for investigating the mathematical opportunities afforded to education students at community colleges, and includes development of a framework for that analysis.


“So I’ve Chosen to Major in Math. Now What?”: Mathematics Students’ Knowledge of Future
Career Options

Katrina Piatek-Jimenez
Central Michigan University
Tim Gutmann
University of New England
1966 - 2007

Research suggests that many mathematics students leave the field of mathematics either during their undergraduate career or shortly after earning a bachelor’s degree in mathematics (Seymour & Hewitt, 1997). Furthermore, preliminary work by the primary author of this paper suggests that many undergraduate mathematics majors view their degree as “limiting” with respect to career options. In this pilot study we investigated what senior mathematics majors plan to do upon graduation, what careers they believe are available to them as mathematics majors, and how they learned about their career options. Such information may be useful for informing the mathematics community on ways to recruit and retain mathematics students.


The Impact of Written Reflections in a Geometry Course for Pre-service Elementary Teachers

Hortensia Soto-Johnon
University of Northern Colorado
RaKissa Dodgen Cribari
University of Colorado Denver
Ann Wheeler
University of Northern Colorado

In this concurrent mixed-methods study we demonstrate how written reflections in a geometry course impact pre-service elementary teachers’ (N = 55) learning and teaching of geometry. Our quantitative data suggest there was not a statistically significant relationship between insightful reflections and task scores. On the other hand, participants performed better on tasks when they participated in written reflections and pre-service teachers, who wrote reflections from the beginning of the semester, produced stronger reflections. We required our participants to reflect and write about what they learned in a discovery based geometry lesson, however our prospective teachers also discussed classroom culture and teaching in mathematics. Our findings indicate incorporating reflections into the mathematics classroom increases student achievement on related tasks, allows pre-service teachers an opportunity to reflect on the learning and teaching of mathematics, and serves as a further assessment of student understanding.



The Assessment of Quantitative Literacy at a Large Public Institution

Yvette Nicole Johnson
Michigan State University
Jennifer Kaplan
Michigan State University

The preliminary results presented here are a response to new developments in quantitative literacy (QL) in the U.S. and, more specifically, at a large, public Midwestern U.S. research university. Moreover, the general perception that a large number of U.S. citizens are underprepared for quantitative tasks in their personal and professional lives as well as other empirical research led a university-wide task force to recommend a curricular shift from an emphasis on traditional mathematical knowledge to a QL focus in mathematics coursework. As the university moves in this direction, our goal is to provide a baseline measure of quantitative literacy for specific groups of students at the university. We will present the findings from a selection of pilot assessment items given to over 500 students. This includes observable differences of the percentage of students who correctly answered a question for various subgroups (pre-college mathematics, pre-calculus mathematics, post-calculus mathematics) being compared.


Impact of Structured Lesson Planning for Adjunct Mathematics Faculty on Classroom Teaching

Matthew G. Jones
California State University - Dominguez Hills
Gwen Brockman
California State University - Dominguez Hills

A professional development (PD) program for adjunct mathematics faculty was designed to train them in developing and implementing lessons that engage students, utilize cooperative or pair learning tasks, and develop students’ conceptual understanding. An early analysis of written lessons submitted as a result of the training revealed that all participants were able to plan engagement tasks, but significant differences were apparent between participants with extensive teaching experiences and those who were relative novices. This paper reports on preliminary findings from classroom observations and surveys of participants and their students, which were designed to measure the extent to which participants’ teaching reflected the learning in the PD.


Students’ Understanding on Eigenvalues and Eigenvectors in Physics Setting and
Implementation of Actor-Oriented Transfer Framework

Gulden Karakok
Oregon State University
Barbara Edwards
Oregon State University

Most of the topics covered in a typical undergraduate Linear Algebra course are often the prerequisite topics for many client disciplines varying from physics, economics, and statistics to various engineering majors. Students are asked to implement ideas from their linear algebra course in their majors. However, the previous studies state that students lack the ability to transfer their knowledge from one course to another. Transfer of learning, traditionally defined as the ability to apply knowledge learned in one context to new contexts. (Bransford et al., 1999; Mestre, 2005) An alternate approach for studying transfer provides a broader perspective on the old definition. Actor-oriented transfer (AOT) approach conceives transfer as the personal construction of similarities between activities where the ‘actors,’ i.e. learners, see situations as being similar. (Lobato, 2003) In the proposed study, researchers first attempt to investigate how students’ understand the concepts of eigenvalues and eigenvectors and then implement the AOT framework.


Undergraduate mathematics tutoring: Concepts or procedures?

Karen Allen Keene
North Carolina State University
Michael Glass
Valparaiso University

We report on beginning research for a new National Science Foundation funded project that studies “cognitive tutoring for conceptual understanding.” In particular, we are interested in answering the question: How can we identify and then develop high quality (expert) tutoring for conceptual understanding of specific solution techniques in differential equations? Using definitions and frameworks for procedural knowledge such as that presented by Star (2005), Hasselbrank and Hodson (2007), NCTM, and NAEP, we offer analyses of ten tutoring sessions for specific solutions techniques in first order differential equations.  This analysis yields an enhanced framework for procedural knowledge and ways to assess it. We also plan to elicit ideas for continuing the study.


The Influence of Symbols on Students’ Problem Solving Goals and Activities

Rachael H. Kenney
North Carolina State University

In this study, the researcher examines the ways in which college pre-calculus students chose activities to perform on a mathematical problem based on what they “see” in the symbolic structure of the problem. The researcher tries to identify students’ goals and activities in problem solving, and tracks the way in which the goals and activities change as the structure of the problem is manipulated. This report is part of a larger dissertation study, and some interesting preliminary analysis results are discussed.


Students’ Notions of Convergence in an Advanced Calculus Course

Jessica Knapp
Pima Community College
Kyeong Hah Roh
Arizona State University

The research literature indicates that the limit concept is typically a difficult concept for students to grasp. However, there is little evidence to indicate how students deal with the mathematical definitions of a limit, specifically the definition of convergent sequence. The purpose of this paper is to examine students’ conceptions of convergent sequences and Cauchy sequences. We examine junior level mathematics students in an advanced calculus course as they prove that a sequence is convergent if and only if it is a Cauchy sequence.


How activity based learning in introductory mathematics courses impacts college
student’s attitude toward mathematics

Ching-chia Ko
Oregon State University
Barbara Edwards
Oregon State University
Gulden Karakok
Oregon State University

This talk describes a nontraditional, activity-based algebra curriculum that was first introduced into a college freshman introductory level mathematics course in the 2006-7 academic year and the attitudes of students toward mathematics and toward the class. The curriculum emphasized conceptual mathematics over procedural skill and encouraged students to actively participate in their own learning. A pre-survey of students’ attitudes toward mathematics was given to all students at the beginning of the term, and again at the end. In fall 2007, we revised our content based partially on the results from these surveys. This year, conceptual mathematics remains the center focus, but procedural skills are also covered. In this talk we will discuss the curriculum change as well as the results of a current survey of students in the fall 2007 class.


Taiwanese Undergraduates’ Proof Performance in the Domain of Continuous Functions

Yi-Yin Ko
University of Wisconsin - Madison
Eric Knuth
University of Wisconsin - Madison
Haw-Yaw Shy
National Changhua University of Education, Taiwan

Recently, a growing number of studies in the United States show that students have difficulty with proofs in advanced mathematics courses (Moore, 1994; Weber, 2001). However, few research studies have specifically focused on undergraduates’ proof performance in the domain of continuous functions annd their abilities to produce proofs and counterexamples, especially research in Taiwan. In this study, we examine Taiwanese undergraduates’ performance constructing proofs and generating counterexamples in the context of continuous functions. While this study is not designed as a comparative study, it will provide results that can be compared with existing empirical studies in the United States. Such comparisons can provide insight into performance differences among undergraduate mathematics students since Taiwanese elementary and secondary school students score consistently high on international mathematical achievement tests. More importantly, our study has broader implications for instructors who would like to improve undergraduates’ proof performance in advanced mathematics courses more generally.


A Trip Through Eigen Land:
Where most roads lead to the direction associated with the largest eigenvalue

Christine Larson
Indiana University & San Diego State University
Michelle Zandieh
Arizona State University
Chris Rasmussen
San Diego State University

An understanding of eigen theory can provide students with powerful ways of analyzing and understanding systemic-level problems in many areas of mathematics, engineering, and sciences. Students struggle to bridge their informal and intuitive ways of thinking with the formalization of concepts in linear algebra (Dorier, Robert, Robinet and Rogalski, 2000; Carlson, 1993). In order to learn more about the interplay of this struggle with students’ learning of eigen theory, a four-week classroom teaching experiment (Cobb, 2000) was conducted during the eigen theory unit in an introductory linear algebra class for university undergraduates during the Fall semester of 2007. Our presentation will consider the relationship between the hypothetical learning trajectory and the learning of two students as it relates to the actual events that transpired within the classroom environment (Simon, 1995).


Emphasizing coordination of measures of center and spread via focus on intervals instead of point values

Hollylynne Lee
North Carolina State University
J. Todd Lee
Elon University

The authors discuss one aspect of the design and preliminary testing of a data analysis and probability module for a technology pedagogy course for preservice mathematics teachers. Through a combination of material focus, simulation/data comparisons and the pervasive use of interval-based activities in lieu of traditional pointestimate exercises, an intended learning trajectory is posited of preservice teachers having a more coordinated view of measures of center and spread.


Exploring the Students’ Conceptions of Mathematical Truth in Mathematical Reasoning

Kosze Lee
Michigan State University
Jack Smith
Michigan State University

The development of students’ mathematical reasoning have generally been examined through their proof schemes and interpretation of logical implications (Hoyles & Küchemann, 2002; Sowder & Harel, 1998). This exploratory study suggests that students’ conceptions of mathematical truth is another important variable in describing their mathematical reasoning. It also explores the relations between their conceptions of truth and their processes of validating mathematical assertions. The analysis of six cases of college students’ interview data and written work suggests that: 1) college students’ conceptions of mathematical truth may not match the normative conception, particularly for the non-math majors; 2) the variations are in graduated continuous steps away from the normative conception; and 3) their processes of reasoning about the truth (or not) of mathematical statements may be influenced by their conceptions of mathematical truth. A conceptual framework of characterizing students’ conception of mathematical truths is also presented as part of the findings.


Constrained by Knowledge: the Case of Infinite Ping-Pong Balls

Amy Mamolo
Simon Fraser University

This report is part of a broader study that investigates university students’ resolutions to paradoxes regarding infinity. It examines two mathematics educators’ conceptions of infinity by means of their engagement with a well-known paradox: the ping-pong ball conundrum. Their efforts to resolve the paradox, as well as a variant of it, invoked instances of cognitive conflict. In one instance, it was the naïve conception of infinity as inexhaustible that conflicted with the formal resolution. However, in another case, expert knowledge resulted in confusion.


A Mathematics Self-Efficacy Questionnaire for College Students

Diana May
University of Georgia
Shawn Glynn
University of Georgia

We are developing a Mathematics Self-Efficacy Questionnaire (MSEQ) that provides college mathematics instructors and mathematics-education researchers with information about students’ self-efficacy (specific confidence) in their ability to learn mathematics. In a pilot administration, students responded to 25 Likert-type items that provided information about students’ self-efficacy in relation to factors such as their gender, previous mathematics achievement, previous mathematics experiences, their use of self-regulation learning strategies, and their perceived level of mathematics anxiety. Preliminary results will be reviewed: The MSEQ data are interpreted using students’ essays and interviews about their mathematics self-efficacy. The findings are viewed in terms of Bandura’s social-cognitive theory of learning, and future research is suggested to refine the MSEQ in terms of its reliability, validity, and convenience of online administration.


Calculus students’ perceptions of graphing calculators and play: Am I ‘doing math’?

Allison McCulloch
North Carolina State University

This paper reports on a qualitative study designed to give voice to the students in the ongoing debate of the use of graphing calculators in calculus. Close attention is given to the students’ perceptions of their mathematical and affective experiences when problem solving in order to answer the following questions: 1) how do calculus students use their graphing calculators to engage playful mathematical activities? and 2) how do calculus students perceive their use of the graphing calculator fits with their perceptions of what it means to ‘do mathematics’? The data indicates that these students’ actions are very much aligned with what mathematicians would define as mathematical problem solving (Polya, 1945; Schoenfeld, 1992), sometimes even mathematical play (Holten et al., 2001). However, these actions do not coincide with the students perceptions of what it means to ‘do math’.


Analysis of Stance in Two Interactive Mathematics Lessons

Vilma Mesa
University of Michigan
Peichin Chang
University of Michigan

This study examined the stance taken by two instructors teaching two mathematics classes for undergraduate students regarding the interplay of two discursive voices, monogloss and heterogloss, used by the instructors. One class was taught under the umbrella of the Emerging Scholars Program (ESP), whereas the other one was intended as a general mathematics requirement for non-science, technology, engineering, or mathematics majors. In spite of the non-ESP class having more instructor-student interactions the ESP class revealed more instances of heterogloss, in which multiple voices were included, acknowledged, and invited. The analysis revealed also the multiple meanings that each voice carried, supporting current views regarding the multi-vocality of interactions. We discuss implications for research and for faculty development regarding managing classroom interaction.


Students’ Ideas About Mathematics (SIAM) and Students’ Ideas About Accounting  (SIAAF):  A Study of qualitative comparison of perceptions held by male and female students enrolled in a first year degree Accounting (AF) course

Sundari Muralidhar
The University of the South Pacific
Nacanieli Rika
The University of the South Pacific

The study was undertaken collaboratively between the Mathematics Learning Support Coordinator and the Coordinator of a first year Accounting (AF) course,   The study was aimed at comparing perceptions held towards Mathematics (M) and Accounting (AF), by male and female students enrolled in a first year degree course in Accounting (AF), and later use the findings to construct a survey for a quantitative study.   The study was undertaken at a multi-modal university which serves many countries.  The subjects were 270 (144F: 126M) students who had enrolled in a service-mathematics course for Social Sciences (MA101), as an academic requirement.   They were a heterogeneous group in terms of culture, academic aptitude and mathematical background.


Students’ Understanding and Use of Representations with Vector Concepts

Sarah Neerings
Arizona State University
Jamie Vergari
Arizona State University

This paper examines four undergraduate students’ understanding of vectors, with an emphasis on the ideas of scalars, span, linear dependence, linear independence, and dimension. This paper also explores the personal concept definitions and concept images held by the students and how their personal concept definition and concept images influenced their understanding of vectors. The students’ understanding of vectors was examined with an emphasis on the students’ ability to explain ideas both graphically and algebraically. The paper also explores what a conceptual understanding of vectors should look like and examines how a students’ procedural understanding or conceptual understanding of vectors affects their ability to make connections. Excerpts from the student interviews demonstrated that the students had mostly procedural knowledge in regards to span, linear dependence, and linear independence, with very few having a connected network of ideas. The results discussed show the importance of teaching conceptually.


Imagining the Imperceptible

Ricardo Nemirovsky
San Diego State University
Michael Smith
San Diego State University

This paper is a preliminary report of research on the nature of mathematical imagination.  The goal of the project is to conduct experimental studies on how mathematicians and students imagine mathematical spaces and events.  This paper focuses on a videotaped interview with a mathematician on topics of linear algebra involving n-dimensional spaces.  The authors conducted a microanalysis of selected segments of the interview focusing on gesture, talk, and symbols drawn on a whiteboard. They chose for this report two episodes of the interview that illustrate hypotheses emerging from the analysis of data.  One hypothesis is about the use of models allowing for the projection of physically unrealizable behavior.  Another hypothesis concerns the use of algebraic expressions as contributing a physiognomy (i.e. a visible appearance expressing fundamental traits of its source) for the symbolized phenomena, even when the latter are postulated to occur in unperceivable spatiotemporal frameworks.


A Local Instruction Theory for Students’ Development of Number Sense

Susan Nickerson
San Diego State University
Ian Whitacre
San Diego State University

Number sense is a widely accepted goal of mathematics instruction. However, teaching with the goal that students develop number sense is challenging. We will present an empirically-tested local instruction theory for students’ development of number sense. Local instruction theories serve to inform the development of hypothetical learning trajectories situated in particular classrooms. We also briefly discuss the associated classroom teaching experiment in a content course for pre-service elementary teachers and the evidence that certain instructional activities led to students’ development of number sense with regard to mental computation. We believe that our local instruction theory is applicable to other mathematics courses.


An Investigation of Graduate Teaching Assistants’ Statistical Knowledge for Teaching

Jennifer Noll
Portland State University

The purpose of this report is to provide a model of statistical knowledge for teaching grounded in an empirical study involving graduate teaching assistants (TAs). Research in statistics education has blossomed over the past two decades, yet there is relatively little research investigating what knowledge is necessary and sufficient to teach statistics well. In addition, despite the fact that TAs play an integral role in undergraduate statistics education, the research community knows very little about their knowledge of statistics and of teaching statistics. In this study, insights into TAs’ knowledge of sampling concepts and their knowledge of student thinking about sampling concepts were gleaned from their ensemi-structured interviews.


First-year Mathematics Majors' Understandings of the Limit Concept
and a Possible New Role for the Concept of Operation

Antonio Olimpio Jr.
Syracuse University

Using the integration of oral language, writing (in natural language) and the CAS MAPLE, I investigated understandings that emerge about the concepts of function, limit, continuity and derivative produced by full-time, first-year mathematics majors from a public university in the state of São Paulo, Brazil. The research, implemented under the guidelines of the interpretive paradigm and of qualitative methodology, was characterized by teaching experiments, which were conducted with eight volunteer participants. The data consisted of individual written answers in natural language and videotapes of the interactions between pairs of participants and MAPLE. In this paper, I present findings regarding difficulties of the participants in dealing with the limit concept and suggest an alternative way, based upon the idea of a generalized mathematical operation, to manage some of these difficulties in teaching practice.


Essential Knowledge of Probability for Prospective Secondary Mathematics Teachers

Irini Papaieronymou
Michigan State University

This preliminary research report attempts to specify the important content topics of probability that should be taught to prospective secondary mathematics teachers in undergraduate probability and statistics courses. In addition, the report aims to identify the aspects of teaching knowledge of these probability topics that should be addressed in these courses. An analysis of a sample of mathematics state standards for grades 6-12, as well as of recommendations from professional organizations, and of three curriculum textbook series is currently under way in the effort to identify this essential knowledge of probability that prospective secondary mathematics teachers need to have.  The ideas presented in this report form part of a larger study currently being conducted by the author in which the aim is to develop a framework for assessing secondary mathematics teachers’ knowledge of probability.


Understanding Iconic Translation

Cassie Pawling
Arizona State University

We see a trend in almost all levels of mathematics that students interpret a situation too literally. In the case of functions, students will graph a function according to what the situation looks like, rather than graphing a relationship between two covarying ob jects, such as distance and time. Conversely, if given a graph of a situation a student with an insufficient understanding may incorrectly simplify a rich relationship to one which resembles the graph. For example, given a graph of speed versus time of two cars, a student with this insufficient understanding may interpret the intersection of the two cars as the point where the two cars meet in space since the curves are laid out spatially. This tendency on the students part to interpret the graph much more literally a picture than it is is termed Iconic Translation, as described by Monk (1992) and Kaput (1992) [4, 3]. I explore the source or cause of such an iconic translation while simultaneously exploring what knowledge allows a student to avoid this overly literal translation. I specifically consider the student’s concept image and definition of function, rate of change, and covariation and the student’s tendency to visually imagine the situation as a means to investigate the correlation between these aspects and the presence of an iconic translation.


A Workshop Based Approach to Calculus Pedagogy

Heath Proskin
California State University - Monterey Bay

The traditional classroom environment leaves little opportunity to encourage students to undertake more challenging multi–part problems. Further, Calculus homework exercises are primarily concerned with developing the skill sets of the students to achieve a level of comfort and proficiency understanding the concepts learned in the classroom. For the last several semesters, I have led weekly, extra-curricular workshops for Calculus students. In these workshops, students work in groups and attempt challenging problems which lie outside the time constraints of the classroom. This research studies the effect of these collaborative workshops on student learning.


A Framework for Interpreting Inquiry-Oriented Teaching

Chris Rasmussen
San Diego State University
Oh Nam Kwon
Seoul National University

In order to improve student learning many teachers, new and experienced, express interest in inquiry-oriented teaching. Such interest is often accompanied with queries regarding the role of a teacher in such classrooms and how inquiry-oriented teachers are able to facilitate classroom discussion in ways that lead to progress on their mathematical goals. The purpose of this report is to contribute to the research agenda on inquiry-oriented teaching by studying one particular teacher in an effort to uncover ways in which he was able to promote his students’ mathematical learning through discourse. In doing so, we offer a framework that characterizes the discursive moves that a teacher can use to create and sustain an inquiry-oriented classroom learning environment. The framework consists of four discursive moves coordinated with five different functions that inquiry serves.


The Emergence of a Complex Graphical Inscription: The Case of a Bifurcation Diagram

Chris Rasmussen
San Diego State University
Michelle Zandieh
Arizona State University

Students throughout mathematics interpret and use graphical inscriptions. Many past studies have examined the difficulties that learners have in interpreting graphical inscriptions and their reluctance to use graphs to solve problems. While these studies add useful insight into student thinking regarding how students’ interpret and use graphs, it leaves open the question of how these graphical inscriptions became mathematical entities for learners in the first place. This purpose of this report is to analyze the emergence (or birth) of one such complex graphical inscription known as a bifurcation diagram. Specifically, we (1) examine the differences in how learners create a bifurcation diagram as compared to how mathematics texts develop the inscription and (2) we examine the bifurcation creation process in terms of conceptual blending.


Investigating the Effectiveness of Technologies Interventions by Assessing Learning Outcomes of Online-College Algebra Undergraduate Students

Atma Sahu
Coppin State University

Integrating technology into the pedagogy is becoming a major part of our educational institutions with the objective to stay competitive in today’s Net Environment. During Fall 2006 to Fall 2008, the investigator taught technology-rich undergraduate mathematics courses that allowed him to immerse his College Algebra students into technology-rich environment. Several other faculty members from the investigator’s campus in various subject areas also introduced new technologies into their course instruction and participated institutions mini-grants program for faculty development Thus, enormous efforts are being made by investigator’s institution academic leaders in the effort to make online learning as effective or perhaps even more effective than traditional on-ground classroom teaching. Since last two years, the investigator had the opportunity to use various synchronous and asynchronous methods to integrate technologies with the delivery of undergraduate mathematics instruction in hybrid-on-ground as well as purely online College Algebra subject area classroom. In this proposal, the investigator proposes to document whether the technology used is easy and has improved student’s learning outcomes in College Algebra classroom. The investigator will present data based teaching and learning outcome findings for on-ground and online College Algebra classes for the duration of Fall 2006 to Fall 2008.


Leveraging teachers’ informal understandings of function for conceiving composition of functions as combining transformations

Luis Saldanha
Portland State University
Sean Larsen
Portland State University

We report on part of an instructional experiment designed to support K-12 teachers in conceiving of the composition of linear functions as combining transformations. Our report will discuss the design and implementation of instructional tasks in terms of a two-phase cycle involving the formulation of an initial and then a revised content-specific instructional theory for the concept of function composition. Moreover, the report will highlight two key interrelated aspects: (1) evidence of teachers’ informal understandings of functions and ways of combining them that emerged as they engaged in a task which provoked them to model a situation in terms of linear input-output relations and the chaining together of such relations; (2) our principled efforts to leverage such understandings in the design of tasks intended to support teachers in formalizing the composition operation.


Drawing conclusions about diagram use in an online help forum

Carla van de Sande
University of Pittsburgh
Gaea Leinhardt
University of Pittsburgh

Free, open (to the public), online homework forums allow students to pose problems from their assignments in a variety of school subjects and receive assistance from volunteer tutors.  We first review our current understanding of calculus tutoring in such forums.  To better understand how this environment affects tutoring, we focus on the use of diagrams in discussions of related rates problems.  Exchanges collected over the course of a year from a representative homework forum resulted in 51 cases in which the mathematical solution would have benefited from the construction of a diagram.  We examined these exchanges for the introduction of a diagram by either the student or tutor(s).  We conclude that students are neither well versed nor confident in the construction of such diagrams and that tutors go to considerable lengths to introduce diagrams as part of the solution process.  This research has implications for calculus instruction and tutoring environments.


Student Learning Using Online Homework in Mathematics

Michael B. Scott
California State University - Monterey Bay

Implementation of online homework and assessment in undergraduate mathematics courses is becoming more common. A natural question to ask is how do such online systems improve or hinder student learning of mathematics? There seems to be little research answering this question. At our institution we use a web-based homework system as a supplement to our Pre-Calculus, Calculus, and Mathematics for Elementary School Teachers courses. The homework system is designed to coincide with the material covered in each course and can be modified if the content changes. We will demonstrate the key features of the system and how students interact with the system. Analysis of the data generated by the system will also be discussed along with what students may or may not actual learn using the system.


The Role of Feelings in Constructing Proofs

Annie Selden
New Mexico State University
John Selden
New Mexico State University
Kerry McKee
New Mexico State University

We describe a perspective and a framework for understanding the role of feelings in proving theorems. We begin with a brief discussion of the nature of feelings. For example, a feeling, such as a feeling of rightness or appropriateness, can express an integrated assessment of more complex activities than could be held in short term memory and assessed consciously. Thus feelings are useful in deciding whether one has written or validated a proof correctly. Also, we see kinds of situations as mentally linked to kinds of feelings that then participate in activating procedural knowledge to yield actions. That is, certain feelings and some parts of procedural knowledge are seen as driving certain aspects of proving. The genesis of the feeling-situation link is illustrated by describing how, for one student, a feeling of appropriateness became linked to a specific aspect of proving.


Consciousness in Enacting Procedural Knowledge

John Selden
New Mexico State University
Annie Selden
New Mexico State University

We describe a perspective for examining the enactment of a common kind of procedural knowledge and how that enactment relates to consciousness. Here, we view procedural knowledge in a very fine-grained way, e.g., considering a single step in procedure, and discuss knowledge that includes, not only how to, but also to, or when to, physically or mentally act. We call the mental structure that links information allowing one to recognize that an act is to be performed, to what is to be done and how to do it, a behavioral schema. We consider how such behavioral schemas might be enacted and how they might interact. The processes associated with a schema’s enaction appear to occur outside of consciousness, but some information triggering its enaction is conscious, and the resulting action is conscious or immediately becomes conscious. We include examples as simple as calculating (10/5) + 7 and mention some implications of this perspective.


The Relationship between Blogging and Students’ Achievement in an Introductory
Undergraduate Mathematics Course

Abbass Sharif
Utah State University

The National Council of Teachers of Mathematics (NCTM) has acknowledged writing as an integral component of mathematics instruction. A number of studies have shown that writing math concepts in prose style increases students’ understanding of mathematical concepts. However, no studies were located that examine the relationship between blogging and the understanding of math concepts. In this study, we are trying to investigate the relationship between blogging and student’s achievement in an introductory undergraduate mathematics course by answering the following two questions: (1) Does blogging enhance the learners’ understanding of a mathematical concept as measured by the students’ school exams? (2) In what ways can a teacher integrate blogging into a math class to improve students’ understanding of math concept?

A Partnership to Promote Inquiry-Based Mathematics Instruction

Tommy Smith
University of Alabama at Birmingham
Bernadette Mullins
Birmingham-Southern College
John Mayer
University of Alabama at Birmingham
mayer@math.uab.edu .
Melanie Shores
University of Alabama at Birmingham
Rachel Cochran
University of Alabama at Birmingham


This study describes the efforts of a mathematics partnership in promoting inquiry-based mathematics instruction and the resulting impact on mathematical knowledge and classroom practices.  The subjects for the study are middle grades teachers and pre-service teachers taking a series of inquiry-based mathematics courses, as well as general university students enrolled in a reformed finite mathematics class.  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 and surveys are used to measure changes in teachers’ instructional practices.  This paper reports the results of changes in participants’ mathematical knowledge and in the instructional practices of in-service teachers.  Implications for changes in other university mathematics courses will be discussed.


Models as tools, especially to help make sense of calculations

Bob Speiser
Brigham Young University
Chuck Walter
Brigham Young University

For us, a model is a tool to solve a problem. Our research concentrates on what learners actually do when they solve problems that demand fresh insight. We want learners to build ideas and understanding they can use to solve new problems. In relation to this goal, the ways specific models help make sense of novel situations can become important subjects for reflection. We anchor our discussion to a concrete example drawn from elementary arithmetic.


Novice College Mathematics Instructors’ Knowledge for Teaching

Bernadette Mendoza-Spencer
University of Northern Colorado
Shandy Hauk
University of Northern Colorado

We report on our analytic inductive analysis of interviews and teaching observations of 6 novice college mathematics instructors (CMIs). Our focus was exploring novice CMIs knowledge for teaching and how it developed in their planning, instructing, and reflecting on instruction. For the first round of interview and observation the focus was on instructor knowledge of student thinking about mathematical concepts in the observed lesson. For the second round we interviewed CMIs about their expectations, grading, and interactions with students around an in-class test.


Linear Algebra Thinking: Embodied, Symbolic and Formal Aspects of Linear Independence  

Sepideh Stewart
The University of Auckland
Michael O. J. Thomas
The University of Auckland

Linear algebra is one of the first advanced mathematics courses that students encounter at university level. The transfer from a primarily procedural or algorithmic school approach to an abstract and formal presentation of concepts through concrete definitions, seems to be creating difficulty for many students who are barely coping with procedural aspects of the subject. In this study we have applied APOS theory, in conjunction to Tall’s three worlds of embodied, symbolic and formal mathematics, to create a framework in order to examine the learning of the linear algebra concept of linear independence by groups of second year university students. The results suggest that students with more representational diversity had more overall understanding of the concept. In particular the embodied introduction of the concept proved a valuable adjunct to their thinking.


Proofs, Purposes and Participation in Undergraduate Mathematics

Sharon Strickland
Michigan State University

This paper examines what aspects of the students in six upper level undergraduate math courses were supposed to change as a result of coursework related to proof (purposes) and roles students played in the courses (participation). From a broad view five of the six courses were lecture or direct teaching, but on closer inspection there were striking differences in their practices. Although the experiences of all students are important, the pedagogic experiences of students becomes extremely important in the cases of preservice secondary mathematics teachers—who are often majors as well. It is hardly controversial to think about undergraduate mathematics classes as sites of teacher (content) preparation. This paper asks what might be learned if we view undergraduate mathematics classes as sites of pedagogical preparation for teachers. Professors have the potential to provide powerful models of teaching although they may not explicitly teach pedagogy. Of the many aspects of pedagogy that might be examined, this paper focuses on the purposes of proof related coursework in six upper-level mathematics and the things students were asked to do as part of the respective course.


Secondary teachers' ways of thinking about exponential functions: An emerging framework and analysis

April D. Strom
Arizona State University

This presentation will focus on an investigation of secondary mathematics teachers' ways of thinking about exponential functions and their development of knowledge as they worked through a collection of exponential activities. A synthesis of the research literature and analysis of the exploratory study revealed that many learners have not developed a robust understanding of exponential functions due to their inability to reason multiplicatively. Furthermore, a review of the relevant literature exposed the lack of a theoretical framework for investigating teachers' understanding of exponential functions. This discussion will present an emerging exponential function framework that served as a lens for analyzing and interpreting the data in this study.



Enhancing Undergraduate Students’ Understanding of Proof

Andreas Stylianides
University of Oxford
Gabriel Stylianides
University of Pittsburgh

Research shows that many mathematics students of all levels of education tend to consider empirical arguments as proofs. Although students’ difficulties with proof are well documented in the literature, the field of mathematics education still lacks knowledge about how to help students overcome their difficulties. This article presents an instructional sequence that we developed over four years of design-research cycles and implemented with promising results in an undergraduate mathematics course, prerequisite for admission to the masters level elementary teaching certification program. The instructional sequence aimed to help students start to develop an understanding of the limitations of empirical arguments and an appreciation of the importance of proof.


Students’ Reasoning about the Concept of Limit and
Entailments of Formalizing the Concept: An Evolving Cognitive Model

Craig Swinyard
Portland State University

The purpose of the ongoing research is 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. Specifically, we aim to: 1) develop insight into students’ reasoning in relation to their engagement in principled instruction designed to support their reinventing the formal definition of limit at a point; and, 2) inform the design of instruction that might support students in reinventing the formal definition of limit.


Developing a “Mathematics for Teachers” course for a new concurrent teacher
education program

Reka Szasz
University of Toronto

The need to provide prospective Mathematics teachers with a strong and teaching specific subject knowledge has been emphasized in many studies recently, and some research has been carried out in order to determine the ideal nature of courses aiming at such knowledge. This talk is about a study on designing such a course for a new concurrent teacher education program. The aim of my research is to develop course content and teaching methods that meet the specific needs of teacher trainees at my university, in order to provide them with an appropriate understanding of Mathematics and a model for teaching it at the same time.



The Ying and Yang of Academic Emotions in Undergraduate Mathematics

Janet M. Thiel
University of Maryland - College Park

How do students feel when they are made to think? My curiosity about academic emotions was peaked by Pekrun’s work, and as a scholar-practitioner, I too believed that there was more emotion to mathematics than the often-studied anxiety. To do further research in this area, for three semesters at the end of the course I asked my students to complete a survey on the emotions they associated with the activities and assignments of the course. These students were generally first-year undergraduates taking a required math class, and they were not math-majors. The assignments included collaborative and individual options, class presentations, on-line and computer assisted practice and assessments, as well as unit assignments of modeling, graphing, and writing. The results showed that positive and negative emotions were often paired by these students, with positive emotions taking predominance.


Cooperative Guided Reflection for Optimization Problem Solving

Kathy Tomlinson
University of Wisconsin - River Falls

This is a study of the ways student learning is impacted by a cooperative guided reflection assignment on optimization problems in Calculus I. The study contributes to an understanding of how the pedagogical practices of writing to learn and cooperative learning effect student growth in problem solving. The investigation uses both quantitative and qualitative methodologies: pre and post surveys of student understanding of problem solving concepts and attitudes about problem solving; comparison of exam performance on optimization problems between students who do the assignment and students in a different section of Calculus I who do not do the assignment; and analysis of students’ written work.


Modeling in a Dynamical System Course

Maria Trigueros
Instituto Tecnológico Autónomo de México

This report is concerned with the development of a research project which integrates APOS and Models and Modeling perspective into the teaching of first order differential equations. A modeling situation was developed and a genetic decomposition for the topic of first order differential equation was developed to guide teacher intervention in the context of an undergraduate course on dynamical systems. Results show that the use of models complemented with a suitable theoretical framework that models students’ construction of knowledge can inform the design of activities to help students reflect on what they know about functions and derivative and to construct a differential equation schema where these concepts are meaningfully related.


Exploration of the role of mathematical discourse in constructing mathematical object

JengJong Tsay
University of Texas – Pan American

Semiotic analysis on mathematical discourse contributes to describing prospective teachers’ construction and communication of mathematical objects. The purpose of this study is to build up a language for use in describing, delivering, and assessing mathematical objects in focus. The data for the study came from observations of three mathematics classes for prospective teachers. A preliminary theoretical analysis using Gray and Tall’s Procept (1994) framework indicates that there are several types of significant discrepancies on participants’ perceptions of signifier-signified-and-referent in mathematical discourse. In this presentation, I will show the types of discrepancy on participants’ perceptions of signifier-signified-and-referent and patterns of negotiation in mathematical discourse when the participants attempted to resolve the discrepancies, successfully or unsuccessfully. Classroom implications and future directions for this study will be discussed.


Making connections between the study of linear algebra content and the study of learning theories

Draga Vidakovic
Georgia State University
Laurel A. Cooley
Brooklyn College - CUNY
William O. Martin
North Dakota State University
Michael Meagher
Brooklyn College/CUNY

This study investigates the impact of the parallel study of learning theory and advanced undergraduate mathematics on prospective and practicing secondary mathematics teachers. Participants at a four-year public, liberal arts college studied learning theories related to mathematics education at the same time they studied advanced undergraduate linear algebra. The researchers investigated how participants use learning theory to gain a deeper understanding of linear algebra and their own learning of content from linear algebra to help make sense of the learning theories. This paper outlines the design and preliminary findings from the project.


An examination of the knowledge needed by a mathematician to teach an inquiry-oriented course in differential equations

Joseph F. Wagner
Xavier University
Natasha M. Speer
Michigan State University

Using case study analysis and a cognitive theoretical orientation, we examine elements of knowledge for teaching needed by a mathematician in his first use of an inquiry-oriented curriculum for an undergraduate course in differential equations. We will present examples of classroom teaching and interview data demonstrating that, despite many years of teaching and while possessing strong content knowledge, mathematicians may still face challenges in changing their teaching practices. Evidence suggests that these challenges result, at least in part, because pedagogical content knowledge acquired through prior teaching practices is not always sufficient to support teachers adopting newer, reform-minded instructional practices. Data such as these, obtained in the absence of questions concerning the instructor’s mathematical content knowledge, highlight other forms of knowledge that are essential to support such teaching. Research such as this is needed to develop support for instructors—especially at a college level—who wish to learn to teach in new ways.


Semantic Warrants, Mathematical Referents, and Creativity in Theory Building

Janet Walter
Brigham Young University
Tara Rosenlof
Brigham Young University
Hope Gerson
Brigham Young University

We examine university honors calculus students’ collaborative development of mathematical methods for finding the volume of a solid of revolution. We qualitatively analyze students’ semantic warrant productions in substantial argumentation during public performances. Students chose specific mathematical referents in the production of solution approaches generated during extended problem solving. Students were convinced of the reasonableness of multiple solution approaches through semantic warrant production during public performances over time and were strongly influenced by the introduction of the First Theorem of Pappus after they invented the theorem in response to mathematical necessity in problem solving. Students’ enactments of personal agency were generative for semantic warrant production and grounded the logical structure of students’ substantial arguments. This study contributes to the literature on the strengths of students’ authentic mathematics creativity within a task-based classroom setting wherein enactments of personal agency are mathematically generative.


How do undergraduates learn about advanced mathematical concepts by reading text?

Keith Weber
Rutgers University

In this paper, two groups—eight strong undergraduate mathematics majors and eight weak mathematics majors—were presented with a standard textbook treatment of a new mathematical concept. The written work they received provided a definition of this concept, examples, theorems and proofs, and homework exercises. The strategies used by each group to learn the concept were recorded, categorized, and compared. Strategies used by the strong mathematics majors included rephrasing the concept definition informally in ways that were meaningful to them and convincing themselves that a theorem was true prior to reading its proof. These strategies were not used by the weaker students.


Abstract Algebra: Proofs and Diagrams

Nissa Yestness
University of Northern Colorado
Hortensia Soto-Johnon
University of Northern Colorado

In this research, we investigate the everyday lived experiences of students’ use of diagrams in developing an understanding of abstract algebra concepts related to groups, subgroups and isomorphisms. Our use of diagrams includes sketches, pictures, illustrations, and gestures. We are particularly interested in how diagrams may assist students with their proof writing abilities and understanding of these abstract algebra concepts. In this heuristic inquiry we collect data in the form of classroom observations, student work and semi-structured interviews. We focus on (a) what prompts students to create or use a diagram, (b) the diagram itself, and (c) how the diagram benefits or hinders the students’ understanding. Through open and axial coding, we identify emergent themes and compare these to existing theories such as those by Gibson (1998).


When Students Prove Statements of the Form (P → Q) ⇒ (R → S)

Michelle Zandieh
Arizona State University
Jessica Knapp
Pima Community College
Kyeong Hah Roh
Arizona State University

We explore the way that students handle proving statements that have the overall structure of a conditional implies a conditional, i.e. (p → q) ⇒ (r → s). Students recruited a proving frame from their experience, which was insufficient for the complexities of the statement. This led them to start with the totality of (p→ q) in ways that were problematic.