2006 Conference on Research in Undergraduate Mathematics Education


Contributed Papers and Preliminary Reports



Reasoning with Time as Parameter in Differential Equations


Karen Allen

Valparaiso University



This paper proposes a characterization of parametric reasoning: how students reason with the dynamic parameter time to support their mathematical activity and deepen their understandings of mathematical concepts. The research was conducted before, during, and after participation in instructional tasks and classroom discourse that focused on understanding and finding solutions to linear systems of differential equations. Students participated in the following identified mathematical practices which characterize parametric reasoning: using the idea that time never stops, reasoning simultaneously in a qualitative and quantitative manner, reasoning by moving from discrete to continuous imaging of time, and reasoning by imagining the motion. Examples of this reasoning are provided from two situated case studies in one classroom teaching experiment. Implications of this research include the possibility that instructional activities can build on student reasoning for students to understand and use the mathematics of change at the middle school, high school, and the university.

Changes in Students' Representations of Outcomes of Experiments

Palma Benko
Passaic County Community College

The learning of probability concepts is very complex (Piaget & Inhelder, 1975; Fischbein, 1975). Garfield and Ahlgren (1988) recommended that longitudinal studies be conducted. This study used qualitative methodology to analyze data in order to describe how a group of students build their probabilistic understanding from grades six, seven and twelve. This paper describes changes in students' representations of outcomes of dice games. I report on the verbal, written, and physical representations of student ideas developed during problem solving sessions. Students produced lists, written mathematical statements, pictures, and graphs. They used their representation to develop convincing solutions to problems posed. This study provides evidence that students did represent their ideas in a meaningful way to understand fundamental probabilistic concepts while working together under conditions of investigating rich tasks, collaborating and justifying their solutions, and having the opportunity to think deeply about their ideas before they receive formal instruction in probability.



Exploring Calculus Students' Understanding of L'Hopital's Rule


Bradley Beauchamp

Illinois State University



It is observed that students often have difficulties in understanding why L'Hopital's Rule works. These difficulties may stem from a shallow understanding of indeterminate forms, functional behavior, and rate of change. This study was designed after the conclusion of a preceding study concerning students' understanding and recognition of indeterminate forms. As such, this study attempts to describe a spectrum of students' conceptions of L'Hopital's Rule and to discover the depth of mathematical connections that are associated with those conceptions. The research questions, and interview protocol, are informed by APOS Theory. Data were collected in the form of written tasks in an interview setting from ten current Calculus II students. The tasks and interview questions were designed to develop a picture of the type of understanding students may have of L'Hopital's Rule and to what degree that understanding is connected to relevant topics in calculus.



On Transcendental Numbers: Teachers' Understanding and Their Practices


Tanya Berezovski

Simon Fraser University



The miraculous powers of modern calculations are due to three inventions: the Arabic Notation, Decimal Fractions and Logarithms (Cajori, 1919). The mathematical concept of a logarithm plays a crucial role in advanced mathematics courses, including calculus, differential equations, number theory, and complex analysis. This report is a part of a larger study on the understanding of logarithms and issues involved in the development of this understanding. Specifically, I focus on in-service high school teachers' understanding of logarithms and how their understanding connects to their teaching of the concept.



Talking Math: Mathematical Language and Explanations

in a Course for Prospective Secondary Mathematics Teachers


Irene Bloom

Arizona State University



High quality mathematical discourse has been advocated for all levels of mathematics. The National Council of Teachers of Mathematics list “Communication” as a major content strand. NCTM also states that is up to the mathematics teachers “to initiate and orchestrate this kind of discourse and use it skillfully to foster student learning”. How do teachers learn to facilitate this kind of discourse when their ability to engage in high quality discourse may be minimal? This presentation reports findings from an ongoing project investigating mathematical knowledge and problem solving abilities in prospective high school mathematics teachers in the context of a capstone content course. In particular, I will characterize the nature of their mathematical explanations and verbal reasoning at the beginning of the intervention and document and describe the changes that occur as the instructor sets classroom expectations and facilitates mathematical discussions.



Students' Reinvention of Straight-Line Solutions

to Systems of Linear Ordinary Differential Equations


Howard Blumenfeld

San Diego State University sdsuhoward@yahoo.com

Chris Rasmussen

San Diego State University



This paper analyzes how students who work in an inquiry-oriented classroom think and develop solution methods for analytic solutions to systems of two linear ordinary differential equations (ODEs). In particular, we seek to understand how students construct and interpret straight-line solutions (SLSs). SLSs are significant mathematical ideas because they serve as the basic building blocks for all other solutions. In particular, SLSs are eigensolutions that span the solution space. This report will also detail the role of proportional reasoning in the process of reinventing SLSs. Our theoretical orientation comes from a socio-constructivist view in which individual math activity is related to emerging classroom meanings and practices. Analysis of student work and justifications suggests that their mathematical understanding of SLSs is a dynamic and evolving entity that matures through classroom discussion and activity. Lastly, this study examines implications for teaching and for the revision of instructional materials.


Learner-Generated Examples:

An inquiry into Students’ Understanding of Linear Algebra


Marianna Bogomolny

Simon Fraser University



Students’ understanding of linear algebra and the effects of different teaching methods on students’ understanding have been investigated from a variety of perspectives. However, these questions have not been examined through the lens of student-generated examples. Since, in order to achieve understanding, students have to be engaged in a mathematical task, the present study investigates whether and in what way the example-generation tasks influence students’ understanding of linear algebra. In particular, the study examines students’ ability to construct examples for mathematical statements and objects in the undergraduate linear algebra course. It aims to analyze and describe what difficulties students encounter when constructing examples, and how example-generation tasks can inform researchers about students’ understanding of linear algebra.



Exploring Mathematical Exploration:

How College Students Pose and Solve Their Own Problems in Classroom Setting


Jinfa Cai

University of Delaware



In the past several years, we have started to investigate the processes that constitute the mathematical exploration processes of solvers.  For example, we examined the mathematical explorations of two college students as they solved an open-ended computer simulation task that involved the path of a Billiard Ball. The results of this study suggest that mathematical exploration can be characterized as a recursive process in which solvers determine goals of action as they formulate their problems, solve the problems, and reflect upon their solution activities to formulate new problems. The results of the study contribute to the development of conceptual frameworks and research tools to capture mathematical exploration processes.  A later study was designed to extend the findings of the earlier study by broadening the scope of the analysis to include new tasks and settings.  This study illustrated how solvers’ cognitive actions can inform their evolving understanding of the task environment, which in turn motivates and enables them to seek out new problems to solve.



Students' Conceptions of Indeterminate Forms


Jim Cottrill

Ohio Dominican University


Bradley Beauchamp

Illinois State University



We observe that students often have difficulties in understanding why L'Hopital's rule works, due possibly to a shallow understanding of indeterminate forms. This research wants to determine whether students recognize indeterminate forms simply through rote memorization or through rich mathematical connections. This study attempts to describe a spectrum of students' conceptions of indeterminate forms and discover what depth of mathematical connections is associated with those conceptions. The research questions are informed by APOS Theory, as was the interview protocol. Data were collected in the form of written tasks in an interview setting from six students in Calculus II. The results indicate students had difficulties due to poor notions of division of numbers. We need to delve into the student's ability to relate division to multiplication. We propose that instructors design activities to reinforce the definition of division of numbers. Specific suggestions in pre-calculus and Calculus I topics will be offered.



Intermediate Algebra Students’ Language-Based Knowledge

and Their Conceptualization of Function


Hamide Dogan-Dunlap

University of Texas at El Paso



I propose to discuss a group of intermediate algebra students’ conceptualization of the concept function when prior knowledge is dominated with common language meanings. Here, the term "common language knowledge/meaning” is used for the meanings that are used outside the context of mathematics. For instance, when “function” is used to describe the role of a teacher, it is considered that the common language meaning is intended. Two intermediate algebra students’ interviews and their responses on a set of post-test questions and definitions are used to argue that existing cognitive structures with nonmathematical origin may have implications for the learning of mathematical concepts.



The Contrast of Hassan’s and Rich’s Understandings

of First Order Differential Equations:

‘Knowing What To Do and Why’ and ‘Rules Without Reasons’


John E. Donovan II

University of Maine



In this presentation results will be reported from a research project designed to investigate two questions: What is the nature of students’ understanding of algebraic and graphical representations of first-order differential equations [FODEs], and what is the nature of students’ understanding of the connection between these representations and their solutions? A series of 4 interviews (1 background, 3 on FODEs) were designed to observe students’ thinking about FODEs within different problem contexts. This talk will focus on two of the students, Hassan and Rich, who came to understand differential equations in very different ways. Hassan had a well defined network of connections for FODEs in which different representations stood for the same thing, and the concept of differential equation was strongly connected with the notion of solution. For Rich, FODEs in and of them selves had very little meaning other than as inputs to procedures.



Student Obstacles and Historical Obstacles to Foundational Concepts of Calculus


Rob Ely

University of Wisconsin--Madison



This study will report on the results from a questionnaire given to university calculus students in September 2005 and again in December, and informed by interviews administered in November. Students will be categorized by the epistemological obstacles they are experiencing to the foundational calculus concepts of limit, function, continuity, and the real number line. Briefly, an epistemological obstacle to a concept is a misconception that is necessarily, or commonly, a part of the concept. Potential epistemological obstacles are drawn from the research of several theories of undergraduate mathematical thinking, including mathematics as metaphor, APOS theory, prototype theory, and concept images. I will compare the combinations of obstacles the students display, and how these combinations change over time, with the combinations and progressions of obstacles found in the historical development of calculus. This should help to illuminate the relationship between the historical structure(s) and the cognitive structure(s) of calculus.



Teaching Innovations for Related Rates Problems in First Semester Calculus


Nicole Engelke

Arizona State University



Related rates problems in first semester calculus are a source of difficulty for many students. These problems require students to be able to visualize the problem situation, attend to the nature of the changing quantities, reconceptualize the variables in a geometric formula as functions of time, and relate these functions either parametrically or through function composition. Being able to successfully solve a related rates problem by engaging in the problem solving behaviors described above relies on the students’ ability to engage in transformational/covariational reasoning. I have developed a sequence of teaching activities that employs a computer program designed to foster the students’ exploration of related rates problems in a covariational context. I am investigating the impact of these activities on students’ abilities to solve related rates problems.



The Calculus Concept Inventory


Jerome Epstein

Polytechnic University



This is a report on the development and validation of a diagnostic instrument, The Calculus Concept Inventory (CCI). It is patterned on the Force Concept Inventory (FCI) in physics, which measures conceptual understanding of the most fundamental concepts of mechanics. The “normalized gain” is widely shown as independent of the entry level of students, of instructor and textbook, but strongly dependent on teaching methodology. It is crucial to determine whether this holds in math as well. The hope to help resolve the “math wars” is to develop a valid and accepted test that can distinguish between methodologies. We report on the process of development, results on initial pilot and first round of field testing, and statistical evaluation of results.



We Know What They Knew, but What do They Know?


Timothy Fukawa-Connelly

The University of Maryland


Steven Jones

The University of Maryland



In our presentation, we propose to describe the daily class activities of a non-traditional abstract algebra class. The description will include both a general description and thick description that was created by observation and video data. As our second contribution to the field, we will offer remarks about the types of mathematical understandings, proficiencies, and habits that the students developed while part of this course. Finally, we will offer a brief report on some these students’ current understandings of group and ring theoretic material (approximately six months after their last course meeting), drawn from a brief written instrument and diagnostic interview.



Students Ideas on Functions of Two Variables


Maria Trigueros Gaisman

Instituto Tecnológico Autónomo de Mexico


Rafael Martínez-Planell

University of Puerto Rico at Mayagüez



The notion of a multivariable function is of fundamental importance in mathematics and its applications. This is a report of a study that uses APOS theory as theoretical framework to study students’ learning about functions of two variables. A very brief introduction to the basics of APOS theory will be presented. It will be followed by a particular genetic decomposition for functions of two variables. An instrument was designed to test the different components of the genetic decomposition. A small group of students was chosen and interviewed using the instrument as a basis. The results were used to improve the instrument and after revision it was then applied to another group of students who were also interviewed. Results from the student interviews show that they have difficulties with the basic aspects of functions of two variables. The results will be discussed, particularly as they relate to the proposed genetic decomposition.


Dialogue as a Practical Tool for Writing Proof


Soheila Gholamazad

Simon Fraser University



This research examines students’ engagement in creating a dialogue as a means towards understanding and creating proofs. In this study I adopt the communicational approach to cognition based on the work of Sfard, according to which thinking is a special case of activity of communication. Participants in this study were pre-service elementary school teachers. I encouraged students to write down the dialogue that they have with/or among themselves while they are thinking and trying to understand or create a proof. The results showed the method of proving through writing dialogue would be practical heuristic for involving students in the process of understanding and creating a mathematical proof.



Do we say what we mean?

Conceptions held by (future) primary school teachers

concerning the concept of angle


Alexandra Gomes

University of Minho



In the process of creating mathematical knowledge, concepts perform a crucial role, being one of the foundations of the mathematical building. Their meaning has to be constructed by the individual in interaction with different contexts, based either on experience or on abstract references. Therefore, definitions are essential in mathematical activity since they constitute a basic component of mathematical knowledge. On the other hand, representations are a fundamental link in the codification and understanding of mathematical concepts. This study will explore the conceptions that (future) primary school teachers have of a basic elementary geometrical concept (angle), especially concerning the role played by definitions and representations.



Network Analysis as Theory Building

for Understanding Graduate Student Communities


Timothy Gutmann

University of New Englaund


Eric Hsu

San Francisco State University



Emiliano Gomez

University of California, Berkeley




Recently professional developers and mathematics education researchers have begun to focus on the needs of mathematics graduate students. This proposal discusses the shortcomings of Lave and Wenger's community of practice model and Nodding's caring-bond model as theories to describe learning within a graduate student community. It proposes social network analysis as a way to empirically generate a working theory of community learning.



Patterns of Change in the Teaching Actions and Behaviors of an In-Service Teacher


Guershon Harel

University of California, San Diego


Osvaldo Soto

University of California, San Diego



The case study reported here examines the teaching practices of one in-service teacher who participated in an on-site professional development (PD) for two years. In this report, we characterize patterns of change in the teacher’s teaching practice over that two year period. Our analysis, focuses on five major categories of change in the teacher’s teaching practices; changes in the way the teacher solicits student ideas, in how she deals with student errors, in capitalizing on solutions offered by students, and in the type of questions she chooses to pose.



No Teacher Left Behind


Shandy Hauk

University of Northern Colorado



Jennifer Kreps

Gustavus Adolphus College jkreps@gustavus.edu


April Brown Judd

Mid-continent Research for Education and Learning


Rhoda Deon

University of Northern Colorado



The article provides results of a study of the pedagogical content knowledge (PCK) of middle school mathematics teachers enrolled in a professional development program. About one-third were already "Highly Qualified" according to No Child Left Behind Act of 2001 criteria and two-thirds were not. Measures of PCK included sub-scores on content, syntactic, anticipatory, and classroom action knowledge from a task-based interview and on a written test based on national standardized tests for teachers. Results indicate that a truly “Highly Qualified” teacher, one with robust PCK, may benefit most from professional development aimed specifically at building and developing self-awareness of mathematically rich anticipatory and classroom action knowledge in addition to opportunities to enrich mathematical content understandings.


Engineering as Bridging Undergraduate and K-12 Mathematics

in the NSF Math and Science Partnerships


Margret A. Hjalmarson

George Mason University



This study presents an analysis of K-12 and undergraduate engineering education initiatives within an NSF grant program. The implications for bridging gaps between K-12 and undergraduate include mathematics and engineering content and curriculum since mathematics courses are a gateway to engineering programs and engineering contexts provide natural venues for mathematics investigations. The types of initiatives include faculty development, K-12 teacher development, curriculum design, and projects designed for recruitment and retention of under-represented groups in engineering.



Patterns of Motivation and Beliefs among Before-Precalculus

College Mathematics Learners


Pete Johnson

Eastern Connecticut State University



This study focused on college mathematics students’ motivation to learn and their beliefs about the nature of mathematics, and on the linkages between these two constructs. Based on the self-determination theory of Deci and Ryan (1985), the literature on belief systems (Green, 1971; Rokeach, 1968), and on reviews of research, five aspects of motivation to learn and six common beliefs about mathematics were considered in this study. Using data from interviews, questionnaires, videotapes, and other documents, three different profiles emerged among the eight students in the study. These profiles were termed “conceptually motivated,” “externally motivated,” and “future value motivated.” Further research is needed to determine the prevalence of these motivational profiles in other college mathematics settings.



What Learner-Generated Examples Reveals about Students’

Understanding of Combinatorial Structures?


Shabnam Kavousian

Langara College/Simon Fraser University



This study investigates students’ understanding of combinatorial structures, and examines their difficulties through an example- generation task. The participants are undergraduate students enrolled in a mathematics course for liberal arts students. The participants were presented with a task in which they were asked to generate an example of a combinatorial problem given a presented combinatorial structure. Their responses were analyzed to examine their difficulties in understanding specific combinatorial structures. As a result of this study, I will suggest some practicable ways to help learners acquire better understanding of this topic.



Mathematical Justification, Parallel Constructions of Knowledge and the Instrumentation Process


Ivy Kidron

Jerusalem College of Technology


Tommy Dreyfus

Tel Aviv University



Processes of knowledge construction about bifurcations of dynamic processes are investigated on the basis of the epistemic actions model for abstraction in context (RBC). The subject is an experienced learner of mathematics. The motivation of achieving a justification for finding the bifurcation points drives her entire learning process. A previous study (Kidron & Dreyfus, 2004) demonstrated that the constructing actions may go on in parallel and interact. The interactions include branching of a new constructing action from an ongoing one, combining or recombining of constructing actions, and interruption and resumption of constructing actions. The analysis showed that the process of justification is related to parallel constructions and in particular, the combining of constructions leads to enlightenment, not in the sense of a formal proof of the statement we want to justify but as an insight into the understanding of the statement. This paper also makes a theoretical contribution to the development of the theory of the RBC model. Specifically, we have analyzed in depth the influence of a particular component of the context, namely an instrument. More specifically in the case at hand, the instrument was a computer tool, Mathematica, and the influence was on the interacting parallel constructions, branching and combining.



Expanding the Participation Metaphor:

What is the Development of Mathematical Practice?


Karen D. King

New York University



This theoretical paper explores the extension of the participation metaphor as outlined by Sfard (1998) to research on undergraduate mathematics education. In particular, by building on the work of Burton (2004) in describing mathematicians’ practices, this paper puts forward for critique a theoretical argument for learning mathematics in undergraduate years as moving closer to the center of the community of practice of research mathematicians. The paper outlines the elaboration of this theory and explores implications for further research and practice.



A Framework to Examine Definition use in Proof


Jessica Knapp

Arizona State University knapp@mathpost.asu.edu


Mathematicians view a definition as having very specific characteristics. Besides defining mathematical objects, definitions give both structure to a proof in a global sense as well as warrants to logical implications in a local argument. Thus definitions play a central role in many proving tasks. Students are required to be able to create and interact with proofs and definitions in a variety of ways. Students are expected not only to produce proofs in homework, but also textbooks and lectures are written with the expectation that students can read and understand proofs and formal definitions. In this talk I propose a framework with which to examine students’ uses of definitions and theorems in proving. The framework is a synthesis of previous literature and research results. I will also seek to illustrate the aspects of the framework using student data from a workshop in advanced calculus.



A Tale of Two Backgrounds: Snapshots of Graduate Students’

Knowledge of Student Thinking in Calculus


David Kung

St. Mary’s College of Maryland dtkung@smcm.edu

Natasha Speer

Michigan State University nmspeer@msu.edu


Improving teachers’ knowledge of student thinking appears to be a successful model for K-12 professional development. For graduate students, with little or no preparation for teaching, opportunities to learn about students’ thinking come primarily on-the-job. We examined knowledge gained by two different graduate student populations early in their teaching careers. While both groups interacted with students during office hours and graded homework and exams, their in-class teaching differed significantly. Some taught traditional recitation sections, mostly answering student questions. Others facilitated Emerging Scholars workshops, writing challenging calculus problems, observing as students work in collaborative groups, and providing Socratic help when necessary. Using a framework adapted from Cognitively Guided Instruction, these “snapshots” show the former group has minimal knowledge of student thinking about key topics in calculus while the latter group has well-developed knowledge of student solution strategies, difficulties, and typical misconceptions.



Impact of Enhancing Reflection and Discourse in College Algebra


Christine Larson

Indiana University


Robert Mayes

West Virginia University



The learning of many post-secondary mathematics students is impeded by their failure to (1) adopt the vocabulary and notations commonly accepted by the mathematics community and (2) recognize fundamental concepts underlying course content. The intent behind constructing a learning environment where students reflect on vocabulary, notations, and concepts and argue about what these things mean and why they are important is to provide the scaffolding for an apprenticeship based model of learning, thus facilitating greater student engagement and learning. This study aims to address the question: Can structurally changing undergraduate math courses to include student reflection, discourse, and inquiry in routine classroom practice improve student learning? The study will target a “high risk” group of students enrolled in the applied algebra course at West Virginia University. On a weekly basis, these students will use journals to reflect on the meaning and significance of what was said (the instructor will facilitate this reflection with pertinent questions). The students will then discuss their reflections in groups of 2-3 people, and one of the groups will meet with the instructor to give him or her insight into class response. This format will continue throughout the semester. Final exam scores will be standardized and the gap between the target group and the control group will be compared with the standardized gap from past semesters, when the target group received supplemental instruction for the same amount of time but the supplemental instruction was of a different nature.



The Examination of Factors Influencing the Integration of Computer

Algebra Systems in University-Level Mathematics Education


Zsolt Lavicza

Cambridge University



Accumulated evidence indicates that the integration of technology into mathematics education is slow compared to the predicted quick spread two decades ago. School-level studies demonstrated that technology integration is highly dependent on teachers' conceptions of technology-assisted teaching, yet, researchers rarely investigate this topic at universities. During the past year, I conducted an exploratory study to examine mathematicians' conceptions of CAS-assisted teaching. I interviewed mathematicians, observed classes, and collected course material at a variety of universities in three countries. In my talk, I plan to share the findings of this study and to highlight the differences between mathematicians' and school teachers' CAS-related conceptions, as well as to discuss the variations in mathematicians' thinking about CAS in the participating countries. In addition, I am developing a large-scale quantitative study to examine CAS integration in university mathematics education programs. Finally, I intend to share and to discuss my preliminary questionnaire design and sampling strategy with the audience.



The Evaluation of the Effectiveness of pre-Freshman Summer Mathematics Courses

on the Academic Performance of Underrepresented Minority Engineering Students


Zsolt Lavicza

Cambridge University


Darryl Koch

University of Michigan



Universities in the Unites States report that it is becoming increasingly difficult to attract sufficient numbers of science and engineering students. Moreover, recruiting and retaining minority students is even a more challenging task. Thus, a number of programs have been established to support minority students before beginning their freshman year. In many programs, adequate mathematics preparedness has been identified as a critical factor for minority students. future success. At our university, various pre-freshman summer programs have been serving minority engineering students for more than 20 years. Our studies indicate that participants in these programs have better graduation rates than their counterparts who do not participate, but little is known about the specific effect of these programs on students. mathematical preparedness. Therefore, we decided to develop a study to examine the effect of pre-freshman summer math courses on students. mathematical preparedness and to assist us to improve these programs. In our talk, we shall outline our research approach and the initial results of our investigation.



Connecting Tasks in Pre-Service Teacher Mathematics Education


Roza Leikin

University of Haifa



This study explores employing connecting tasks in mathematics education of pre-service mathematics teachers (PMTs). This report presents on-going study with PMTs, which is based on the findings of our longitudinal study with in-service mathematics teachers. We found that in-service teachers' mathematical and pedagogical knowledge is curricular oriented and mostly craft and prescribed. We found that teachers' rarely remind systematic sources of knowledge associated with connecting tasks. In this research project we explore development of PMTs' knowledge in systematic mode. The research is performed in the context of geometry course focusing problem solving in different ways. It employs observations as well as written questionnaires. The 3D model of teacher knowledge is applied for the analysis of PMTs' knowledge development.



Building an Understanding of Student Use of Graphing Calculators

as a Tool for Problem Solving


Allison McCulloch

Rutgers University



Graphing calculator use plays a significant role in calculus instruction and assessment. The purpose of this study is to begin to construct an understanding of how calculus students use the graphing calculator as a tool when they are working independently and how they feel its use influences their problem solving experiences. Fourteen students participated in a task-based interview. Four of these students participated in a follow-up interview during which they reflected on the tasks and their feelings about calculator use. The results of this study show that students are often triggered to use the calculator to find a new approach for solving a problem. This new approach is often a visual one. Additionally, there were sex differences in student assignments of authority.



A Continued Look at the Impact of Web-Based Linear Algebra Tools

on Understanding


David E. Meel

Bowling Green State University



This talk will outline a variety of web-based linear algebra tools which focus on concepts such as linear transformations, eigenvalues and eigenvectors, change of bases, matrix multiplication, diagonalization, etc. By utilizing a variety of data collection methods, such as journal entries, written responses, concept mapping activities and interview questions, the talk will discuss how the students interacted with the web-based tools and what connections they built. Indications point to changes in student descriptions and connections they made as a result of interacting with the web-based tools. Samples of student work and statements concerning the use of such tools will be presented.



Is this the answer? Is this what I was supposed to do?

Control Structures in Introductory Calculus Textbooks


Vilma Mesa

University of Michigan



Preliminary report of an analysis of 9 introductory calculus textbooks that looked at strategies available to students for (1) determining whether an action was relevant when solving a problem and (2) deciding that the problem was solved. This is known as the control structure, and it plays an important role in using validation strategies in mathematics. The books differ in terms of audience (Honors, Regular, Applied, Science & Engineering) and in terms of influence by the calculus reform movement. The study describes the control strategies that are present in lessons and examples associated with the derivative (~800 examples) and establishes differences across textbooks. There are two main hypotheses in this study: (1) Strategies tend to rely more on content at stake in books intended for honors students, and (2) Reform oriented textbooks provide the control structure more explicitly. Results have implications for research, teaching assistant training, and teaching.



Learning What Needs to be Justified in Mathematical Proof


Susan Nickerson

San Diego State University


Chris Rasmussen

San Diego State University



The overall goal of this preliminary report is to investigate undergraduate students' epistemological shifts in their mathematical justification and to identify pedagogical factors that foster and/or constrain critical shifts in students' ability to create mathematical arguments. We view proof as a social process in which participants of a learning community move from peripheral participation to more full members of a community engaged in the mathematical activity of proving. This report analyzes the impact on students of an instructional intervention that modifies of the classical format of a two-column proof which reifies the presentation of a proof and the questions asked by the writer or a reader of the proof.



Model Analysis in Undergraduate Mathematics Education Research


Michael Oehrtman

Arizona State University



This talk will outline the data analysis technique of model analysis theory and its potential applications to mathematics education research. Pioneered in physics education, this theory uses basic principles of quantum mechanics as an inspiration for accounting for non-deterministic, multi-state student understandings. We apply the techniques to two conceptual areas in mathematics education, understanding of functions and limits. Model analysis of 76 secondary mathematics and science teachers participating in a professional development focusing on function concepts reveals little change in the model structures related to action and process views. Analysis of 42 students in standard undergraduate calculus courses reveals a slightly increased tendency of students to apply a wider variety of models, including incorrect models, at the end of the course. Students in an experimental course designed to foster particular conceptual models for limits, exhibited increased tendencies to choose these models and decreased tendencies to choose incorrect versions of them.



Considering the Norms and Practices of Mathematicians

when Analyzing an Advanced Undergraduate Mathematics Classroom


Katrina Piatek-Jimenez

Central Michigan University



Just as social norms, sociomathematical norms, and mathematical practices develop within a classroom community (Cobb & Yackel, 1996), such norms and practices have also developed within the community of mathematicians. These norms and practices influence the norms and practices within a classroom through the instructor, textbooks, and curriculum. As a result, when analyzing an advanced undergraduate mathematics classroom, it is important to consider the role that the norms and practices of the mathematics community play in the development of those in the classroom. During this talk I will present a proposed framework based on the Emergent Perspective that will also incorporate the influence of the community of mathematicians into the study of undergraduate mathematics classrooms. This framework is grounded in qualitative data collected during the study of two “Introduction to Proofs” classes. Data from this study will also be discussed in the context of the framework.



Developing Deep, Connected, and Durable Understanding of Combinatorics


Arthur B. Powell

Rutgers University



This study, part of a longitudinal investigation into the development of students’ mathematical ideas and reasoning, examines the deep, connected and durable understanding of combinations evidenced in students’ reasoning through transitivity to establish an isomorphism between contextually different problems. A goal of teaching combinatorics is to have students understand computational techniques of combinatorics and combinatorial notations as well as to use flexibly these techniques and notations to solve varied problems. However, mathematics educators are uncertain about how to achieve this goal. Moreover, the question is under-researched. The results of this study suggest that learners require time to enable ideas to dawn and mature, that problems that invite students to think deeply and discursively enhance their mathematical understanding, and that strand of varied, spiraling problems help students develop deep, connected, and durable understanding and facility with basic combinatorial ideas.



Effects of DNR-Based Instruction on the Teaching Behaviors of Algebra Teachers


Jeffrey M. Rabin

University of California San Diego


Guershon Harel

University of California San Diego



We report further results from an ongoing study of changes in the teaching behaviors and knowledge base of algebra teachers following professional development institutes embodying "DNR-based instruction", a conceptual framework for the learning and teaching of mathematics. Teaching behaviors of two teachers are categorized and evaluated for alignment with DNR. Probable effects of these behaviors on student learning are conjectured. We document changes in teachers' practice following their experience of DNR-based instruction. The choice and effectiveness of teaching behaviors depend on adequate mathematical and pedagogical content knowledge.



Videocases as a Didactical Intervention in an "Introduction to Proof" Course


Manya Raman

Rutgers University



This preliminary report describes how videocases can be used to help students overcome difficulties in learning to produce mathematical proofs. The study was conducted in the context of a bridge-type proof course at the university level. Students that had particular difficulty writing proofs were asked to volunteer for an extra help session. In this session, students watched and discussed videos of "experts" talking aloud while proving several of the proofs that were difficult for the students. The goal of the sessions was to help students connect informal and formal aspects of proof. In this talk, we discuss the relative advantages and disadvantages of this intervention, and we solicit ideas for improvement for the second iteration.



Students’ Proofs for the Shapes of Graphs of Solutions in the Phase Plane


Chris Rasmussen

San Diego State Univeristy chrisraz@sciences.sdsu.edu

Jill Nelipovich

San Diego State University jnelipov@ucsd.edu


Bohdan Rhodehamel

San Diego State University bohdanr561@yahoo.com



This paper reports on student proofs for the shapes of graphs of solutions in the phase plane for systems of two linear homogenous differential equations with constant coefficients. We define proof to be an argument and we draw on Toulmin’s scheme for the anatomy of an argument. Data for this analysis comes from an eight-week teaching experiment conducted in an undergraduate differential equations course in a large southwestern university. We found that students’ proofs relied on six sources (data) for the basis of their claims. In our presentation we will illustrate and clarify the students’ deductive arguments for solution graph shapes. Findings from our research contribute to contemporary characterizations of proof as a social process. The presentation will conclude with implications for teaching and instructional design.



Association of Students' Conception of Limits with Their Levels of Reversibility in the Context of Sequences


Kyeong Hah Roh

Arizona State University



This study explored how students conceptualize the relation between ε and N in the definition of the limit of a sequence. Students participated in the ε–strip activity which was specially developed to describe the relation between ε and N. This study results that students' understanding of the relation between ε and N fell into 5 major levels. In the classification of levels, it was regarded as an important factor whether or not students understand the arbitrary choice of ε, the dynamic feature of ε to decrease to 0, and the dependency of N on ε. It was also found that students’ levels of reversibility were associated with their understanding of the limit of a sequence. Actually, throughout the ε–strip activity, students’ reversibility and/or their conception of limit had been improved. It implies the development of reversibility should be considered in teaching limits of sequences.



Student Understanding of Accumulation and Riemann Sums


Vicki Sealey

Arizona State University


Michael Oehrtman

Arizona State University



Student understanding of the conceptual structure of the Riemann sum definition of a definite integral is a topic on which little research has been done. This research uses the ideas of approximation (finding over and under estimates, determining a bound on the error, and finding an approximation accurate to within any predetermined bound) to develop a strong conceptual understanding of accumulation in college calculus students. Data shows that students with a strong understanding of Riemann sums are much more successful at setting up nontrivial definite integrals and solving nontrivial accumulation problems.



Students’ Strategies for Constructing Mathematical Proofs

in a Problem-Based Undergraduate Course


Jennifer Christian Smith

The University of Texas at Austin


Sera Yoo

The University of Texas at Austin



In order to more closely examine the strategies and processes students employ when constructing mathematical proofs, we followed six students enrolled in a problem-based undergraduate number theory course. During a series of interviews, the students were asked to construct proofs of various number theory statements. The students’ strategies for constructing proofs varied depending on context, but in each case they were primarily engaged in making sense of the mathematics, rather than applying a previously known proof strategy. In addition, their strategies varied greatly from one proof to another; they did not appear to show a preference for a particular approach to constructing a proof. We claim that the problem-based structure of the course facilitated the development of these students’ relatively fluid approaches to proof construction.



Undergraduate Mathematics Teachers' Knowledge of Students' Strategies

and Difficulties with Derivative Problems


Natasha Speer

Michigan State University


Sharon Strickland

Michigan State University



Nicole Johnson

Michigan State University


Beste Gucler

Michigan State University



Teachers’ knowledge shapes decisions they make while planning instruction and while teaching. Knowledge of student thinking (e.g., typical solution paths, difficulties) appears to be an especially powerful influence on teachers’ practices and a productive site for professional development. These issues, however, have not been examined in many content domains. This research examined college mathematics teachers’ knowledge of student thinking about derivative and concepts that interact with learning derivative (e.g., function, limit). We analyzed participants' knowledge of student strategies and difficulties for tasks previously used in research on student thinking. Comparisons are made between participants' knowledge of student thinking and findings from research on student thinking. Analyses indicate that participants did not possess rich knowledge of student thinking. Specifically, most struggled to generate solution paths other than the ones they had used and anticipated only procedural difficulties. Implications for practice and design of professional development are discussed.


Students' Use of Representations to Develop Ideas in Combinatorics


Lynn D. Tarlow

City College of the City University of New York



This study documents the mathematical development of a group of students who built representations to solve challenging combinatorics tasks and then refined and linked those representations in order to develop an understanding of the relationship between the tasks, the combinatorial idea of (m choose n), and Pascal's Triangle. The students' use of representations was critical for their development of an understanding of combinatorics, a topic that has been noted to cause difficulty for college students. This suggests that ideas of combinatorial notation, perhaps, should not be imposed on students, but rather students should be given time to develop these ideas in environments that encourage them to recall, produce, refine, and connect representations.


Dialogue Based Teaching Practice in an Undergraduate Analysis Course


Stine Timmermann
Roskilde University, Denmark

This is a report of the results of a pilot study of an undergraduate analysis course at a Danish University, characterised as a reform university. The specific course was examined as a part of a larger project concerning the establishment of plausible connections between teaching practices and students’ problem solving strategies and reasoning. Data was collected in the form of classroom observations based on a pre-developed schedule, semi-structured interviews with students and teacher and monitored problem solving sessions with students grouped in pairs. The results reveal a discrepancy between the expressed needs of the students (highly influenced by their disabilities to solve the assigned tasks), the teacher’s intentions and the actual behaviour of students and teacher in class. Possible explanations for the course of events and guidelines for further improvements are proposed.


Building Mathematical Connections through Communications


Elizabeth B. Uptegrove

Felician College


Carolyn A. Maher

Rutgers University



We describe here a problem-solving session where, through collaboration and discussion, a group of students strengthened their understanding of some features of Pascal's Triangle by identifying connections between two isomorphic combinatorics problems with different surface features. The question that guides our analysis was: How did communication among students in a problem-solving group contribute to the group's success in making sense of problems in combinatorics? We use data from a longitudinal study that has followed a group of public school students from first grade through college. Data for this analysis are taken from an after-school problem-investigation session involving four high school students. The theoretical framework is provided by Maher and Sfard, who stress the importance of communication in mathematical thought. Our view, supported by our analysis of the students' conversation, is that discussing mathematical ideas and communicating about them are important prerequisites for recognizing connections between problems of equivalent structure.



Spontaneous Online Discussions in Mathematics


Carla van de Sande

University of Pittsburgh



Spontaneous online discussion boards (ODBs) are forums where students post questions, receive explanations, and discuss both conceptual and procedural mathematical issues. In contrast to ODBs affiliated with a course or institution, spontaneous ODBs involve anonymous participants who do not have contact outside of their online discussions and who participate in discussions that are generated by queries originating from a wide variety of contexts. Spontaneous ODBs may be one of the few resources outside of school available to upper level mathematics students and appear to be a popular source of tutoring assistance. Although tutoring has been studied in other contexts (e.g. face-to-face) and in course-initiated ODBs, it has not been studied in the context of spontaneous ODBs. This research investigates tutoring exchanges in spontaneous ODBs on advanced mathematical topics such as limit and seeks to characterize the associated discourse community in terms of pragmatics and practices of instructional explanation.



Student-to-Student Questioning: A Case for Autonomous Scaffolding


Janet G. Walter

Brigham Young University



This paper examines student-to-student questioning and problem posing within a framework of personal agency, purposeful choice and learner-initiated inquiry in problem solving. This qualitative study is part of a precalculus strand in a larger longitudinal study of students’ mathematical thinking. Six high-school students worked collaboratively on a task that suggested the use of polar coordinates, a topic with which these students were unfamiliar. Questions emerged as essential features of students’ problem-solving activities. Students asked questions that were linguistic representations of mathematical thinking and purposed toward making sense of a problem situation. Implications for teaching practice will be discussed.



How do Undergraduates Complete Homework in Abstract Algebra

and What Do They Learn From Doing It?


Keith Weber

Rutgers University



Three undergraduates were videotaped thinking aloud while completing homework assignments for two one hour sessions. Afterwards, they were asked what they learned by completing these exercises as well as more general questions about their strategies for completing their homework and their learning in general. The results of this study reveal the different strategies undergraduates use to complete their homework and how these strategies significantly affect what they learn from doing their homework.



Developing Undergraduates' Understanding of Trigonometric Functions

in a College Geomtery Course


Keith Weber

Rutgers University



In this talk, I will define what it means to understand a trigonometric function, using Gray and Tall's (1994) notion of procept. I will describe instruction based on this theoretical analysis. I will then report the results of a comparison study



Reconceptualizing Mathematical Objects as Mediating Discursive Metaphors


Aaron Weinberg

Ithaca College



There are several frameworks used to describe students’ conceptions of mathematical entities as processes and objects from a Piagetian perspective. The work of Vygotsky and Bakhtin suggests focusing on the use of tools such as symbols and language that mediate thinking and discourse. Using these ideas, we present a new framework for describing functions as mathematical objects in discourse. Incorporating the ideas of facet, object-mediated discourse and a mediational toolkit, this framework is used to analyze undergraduate calculus students' discourse about functions. The components of the framework describe the metaphors students use to talk about and work with functions along with the focus of their discourse. The two framework components allow us to analyze different aspects of discursive mathematical objects, and suggest an instructional focus on developing discursive competencies as students participate in mathematical activity.



Effects of Highly Contextualized Learning Activities on Students’

Mathematical and non-Mathematical Recall


Dale Winter

University of Michigan



Many claims have been made concerning the advantages of embedding mathematical problems in “real world” contexts. One specific claim is that learning mathematics by solving such problems may enhance students’ recall of mathematical concepts and procedures (Blumberg et al., 2005; Taylor, 2001). This report describes the results of a study testing this hypothesis within introductory undergraduate mathematics courses (precalculus and calculus). The study showed that the use of elaborate contexts significantly enhanced students’ recall of the context, compared to problems that were embedded in routine contexts. When mathematical concepts or procedures were ones that the students had worked with frequently, the use of elaborate contexts produced no significant gains in students’ recall of the mathematics. However, when the concepts or procedures were ones that students had relatively little exposure to in the course, the use of elaborate contexts was associated with significantly enhanced student recall of mathematical concepts and techniques.

Learner Generated Examples: From Pedagogical Tool to Research Tool


Rina Zazkis

Simon Fraser University



Watson and Mason (2005) make a compelling argument for the use of learner generated examples as an instructional tool. I wish to explore whether and how learner generated examples can be used as a research tool, that is, as a lens for researcher to interpret students' knowledge or understanding of subject matter. I will consider series of learner generated examples and invite inferences about the students. Further, among different perceptions on learning I will examine a possible view of learning as "enriching individual's example space".