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


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

Curtis Bennett Loyola Marymount University cbennett@lmu.edu

Jacqueline Dewar Loyola Marymount University jdewar@lmu.edu

This article presents the results of a study of the evolution of students’ understanding of proof as they progress through the mathematics major curriculum at a medium-sized comprehensive university. The study attempted to identify which courses or learning experiences promote growth in student understanding of proof. The researchers employed surveys and think-alouds on proof. As an aid in analyzing thirteen think-alouds on proof (12 students, 1 faculty), the researchers developed a typology of mathematical knowledge that includes six cognitive and two affective components. Then the typology was expanded into a taxonomy that describes the journey toward proficiency in each of these components. The resulting mathematical knowledge-expertise taxonomy can be applied to analyze student work and instructor responses.

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

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

Susan S. Gray University of New England Biddeford, Maine sgray@une.edu

Barbara J. Loud Regis College Weston, Massachusetts barbara.loud@regiscollege.edu

Carole P. Sokolowski Merrimack College No. Andover, Massachusetts carole.sokolowski@merrimack.edu

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

Jon Hasenbank University of Wisconsin - La Crosse hasenban.jon@uwlax.edu

Ted Hodgson Montana State University hodgson@montana.edu

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

BEST PAPER AWARD, 2007 CONFERENCE

Teaching Assistants Learning to Teach: Recasting Early Teaching Experiences as Rich Learning Opportunities David Kung St. Mary's College of Maryland dtkung@smcm.edu Natasha Speer Michigan State University nmspeer@msu.edu Just as doing mathematics creates opportunities to learn mathematics, “doing teaching” creates opportunities to learn to teach. Nowhere is this more applicable than for graduate students who have little or no teaching training prior to their first teaching assignments. We report on our analysis of how the research literature on teachers’ on-the-job learning can be applied to the context of graduate student professional development. We combine this analysis with our synthesis of findings about the role of teachers’ knowledge about student thinking in shaping instructional practices and student learning opportunities. Our findings take the form of a framework, grounded in research on teacher learning, to guide the design of activities andprograms to equip graduate students with the skills and dispositions to inquire into and learn from their teaching experiences.

Elise Lockwood Portland State University elockwoo@pdx.edu

Craig Swinyard Portland State University swinyard@pdx.edu

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

Ami Mamolo Simon Fraser University amamolo@sfu.ca

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

Bernadette Mullins Birmingham-Southern College bmullins@bsc.edu

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


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


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


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


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


This presentation will relate to a range of contemporary theoretical perspectives by presenting a framework of three modes of thinking that operate so differently as to present essentially three distinct ‘worlds of mathematics’—conceptual embodiment, proceptual symbolism and axiomatic formalism—which will be abbreviated in the context of this theoretical framework to embodiment, symbolism and formalism. Human learning will be addressed in terms of compression of knowledge in which important aspects of a situation are named and built into rich thinkable concepts that enable thinking to be performed by making connections between the thinkable concepts to build successively more sophisticated conceptual structures. In this way, any new concept needs to be seen in the light of previous experience, building on aspects within the individual’s concept image that I call met-befores. Symbolism builds on earlier embodied and symbolic met-befores; formalism builds on earlier embodied, symbolic and formal met-befores. All three operate with ongoing interchange between them.

To illustrate the power of embodiment to underpin formal thinking, the embodied notion of local straightness will be contrasted with the symbolic notion of local linearity to show that while the latter has great computational power, it lacks the embodied meanings that lead naturally to significant formal meanings in mathematical analysis. On the other hand, proceptual computations in many areas such as groups, vector spaces, mathematical analysis will be shown to lead not only to formal definitions but also to structure theorems that provide deeply meaningful embodiments.

The presentation will consider a range of recent studies which show how embodied met-befores can both enhance and impede formal thinking and discuss how a combination of (conceptual) embodiment, (proceptual) symbolism and (axiomatic) formalism relates to the all-embracing Lakoffian concept of embodiment, and process-object encapsulation that is the main focus of APOS theory.

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


Seven papers, spread over two sessions, will report on the first 18 months of a five-year project that aims to improve secondary mathematics teachers’ instruction while at the same time producing theoretical understandings of the intervention’s effects on teachers beliefs, values, and knowledge. One paper will give an overview of the project and a broad summary of its results. Five other papers will report on the effects of covariational approaches to functions on teachers’ reasoning, on making meaning as a means for professional development, on decentering as a critical component of teacher practice, on teaching for meaning in trigonometry, and on an extension of the project into one classroom.


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

Ian Whitacre San Diego State University ianwhitacre@yahoo.com

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


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



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


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


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

Axelle P. Faughn California State University Bakersfield afaughn@csub.edu

Terran Felter Murphy California State University Bakersfield tfelter@csub.edu

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