Lara Alcock University of Essex lalcock@essex.ac.uk

Keith Weber Rutgers University khweber@rci.rutgers.edu

Anne Seery Rutgers University aseery85@eden.rutgers.edu

This report will be about a DVD designed to generate discussion between mathematicians about their students’ thinking.  It shows students from transition course working on proof problems.  Half of the screen shows the student and the other half shows subtitles and their writing.  Segments of video are interleaved with screens of questions designed to prompt reflection on issues in mathematics education such as students’: lack of appreciation for the role of definitions in formal mathematics (e.g. Vinner, 1991); restricted knowledge of examples (e.g. Moore, 1994); lack of proving strategies (e.g. Weber, 2001). The availability of this DVD will be timely in the UK context in particular, since there is an increased emphasis on teaching qualifications for new lecturers but a lack of content-based materials for subjects such as mathematics.  In the presentation we will show segments of the video and present early feedback from mathematicians at a number of universities.

Karen Allen Valparaiso University karen.allen@valpo.edu

Advancing Mathematical Activity is a new construct that allows mathematics educators to study students’ participation in the K-16 mathematics classroom.  Rasmussen, Zandieh, King, and Teppo (2005) examined students’ participation in mathematical activity as it advances in sophistication as an alternative to Tall’s notion of advanced mathematical activity (1994). By using the concept of advancing mathematical activity, I interpret students’ mathematical activity in one inquiry oriented differential equations class and use the interpretation to evaluate their growing participation in the mathematical community of practice (Wenger, 1998). The construct of advancing mathematical activity identifies five types of activity that students may participate in: symbolizing, algorithmatizing, defining, justifying, and experimenting.  The analysis of the participation in the classroom yielded students enacting all but the defining activity. This kind of analysis provides a better understanding of the mathematical activity of student mathematicians developing at the university level.

Jason K. Belnap Brigham Young University belnap@mathed.byu.edu

Kimberly Allred Brigham Young University kim.allred@gmail.com

Recently there has been a growing focus on understanding Graduate Mathematics Teaching Assistants (GMTAs), their teaching development, and preparation; however, we know little about our GMTA population and how mathematics departments are using them. Addressing this problem, we conducted a nationwide, electronic, survey study of U.S. mathematics departments, to determine the extent of GMTAs’ instructional involvement. Findings reveal important di_erences between GMTAs’ responsibilities and GTAs’ in other disciplines. Further, they show diversity among mathematics departments’ use of GMTAs, particularly in three areas: GMTA Population Size, Cultural Diversity, and Autonomy. Departmental variations along these variables can create different social contexts, plausibly impacting GMTAs’ professional development and social networking opportunities. This report will detail and support these findings, answering the question, “What is the extent and nature of GMTA involvement in college mathematics instruction?”

Jason Belnap Brigham Young University belnap@mathed.byu.edu

Preparing graduate students to teach can directly impact the quality of college mathematics instruction. Many programs have been developed to help Graduate Mathematics Teaching Assistants (GMTAs) be successful, but almost no research underlies most of these programs. Little has been done to understand these programs’ impact and structure. September 2006, we initiated a longitudinal research study, aimed at developing and understanding one form of professional development: the use of Video Observations with Peer-feedback Sessions (VOPS). Utilizing qualitative methods, we seek to understand how VOPS can be structured to foster social teaching networks and what variables impact its implementation. Preliminary data highlight factors impacting session discourse (e.g. teaching assignments and prior teaching experience); they further suggest that VOPS works in small group settings and can effectively engage GMTAs with prior teaching experience or diverse teaching assignments. In this report, I will detail these findings and discuss preliminary implications for such a program’s implementation.


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 workshop will provide techniques for maximizing an individual's use of the new Math Gateway and the Math Digital Library provided by the MAA for all members.  MAA's Math Gateway is a remarkable search portal designed to implement searching undergraduate mathematics materials within the National Science Digital Library (NSDL). The Math Gateway brings together collections with significant mathematical content and services of particular importance to the delivery and use of mathematics on the Web. Math Gateway also provides a home page for the user that not only keeps track of favorite resources, but allows for a discussion board and shared resources with selected individuals. The workshop will show the particulars of using the Gateway and support will be provided from the makers of the Gateway.


This study is a contribution to the ongoing research in undergraduate mathematics education. It focuses on example-generation tasks as a methodology to probe students’ understanding of mathematics. It is guided by the belief that better understanding of students’ difficulties leads to improved instructional methods. The study introduces example-generation tasks as an effective data collection tool to investigate students’ learning of mathematical concepts. In particular, this study focuses on students’ understanding of the concepts of linear algebra and pre-service and in-service teacher understanding of the concepts of transcendental numbers. Simultaneously, it enhances the teaching of mathematics by developing a set of example-generation tasks that are a valuable addition to the undergraduate mathematics education.


In this talk we will explore findings from the Whole Number Study. These findings indicate that teachers may incorporate reform approaches while maintaining particular beliefs or ideas about mathematics, even when the pedagogical implications of these beliefs and ideas interfere with student learning goals, as espoused in reform curricula. To illustrate this finding, we will examine a particular mathematical idea that 2nd grade teachers in our study enacted as a “mathematical rule.” This rule, which we will refer to as the “right to left” rule (RL-rule), manifests itself when students are asked or required to operate on numbers from right to left (i.e., ones, tens, etc.). Having examined teachers’ use and views on the RL-rule, we will then turn our attention to the implications of this work for preservice teacher (PST) mathematics content courses and the ways in which Lesson Reviews might facilitate the exploration and examination of PSTs’ ideas and beliefs about mathematics. 

This paper describes the actions taken by a teacher in a differential equations course who set out to develop a classroom culture characterized by inquiry. One of the primary means for achieving this goal was to allow for and even request, and then to investigate student-generated conjectures. The data shows that a culture of inquiry was developed where students freely communicated and then explored their own conjectures in the classroom. Moreover, as a social norm, the nature of the students’ conjecturing activity evolved over the course of the 16-week semester. The data makes it clear that the teacher played a pivotal role in both the initial negotiation of the social norm of conjecturing and its subsequent evolution. The results of this research help to further clarify and underscore the important role which the teacher plays in instigating reform in mathematics education.


This talk will detail preliminary research conducted in the West Virginia University (WVU) Liberal Arts Mathematics course.  Initial research centers on developing an assessment to be used as a pretest and posttest, the focus of which is using mathematics to solve everyday problems.  Further research involves the effect of an interactive computer laboratory component on student performance in and satisfaction with the course.  The control section will meet for two lectures on core material and one on applications per week.  A second section will have two lectures on core material and one computer laboratory meeting a week, in which students interactively explore applications using technology.  The two sections will be compared using the pre/post assessment, a pre/post attitude survey, exams, quizzes, Personal Response System questions, and attendance.  Matched pairs of students will also be compared using the measures listed above, to control for possible differences between the two sections.


In this project we investigated the interactions among members of a professional learning community (PLC) of secondary mathematics and science teachers. We report on our investigations of the facilitator’s role in promoting meaningful discourse among the learning community participants. We describe meaningful discourse in this PLC context as involving substantive conversations about aspects of knowing, learning and teaching mathematics content. As our research has evolved we recognized that facilitators who made efforts to understand the thinking and perspective of the PLC members were better able to engage the members of the community in meaningful conversations. We call this form of engagement acts of decentering. The data revealed five manifestations of decentering. We illustrate through vides how four different PLC  facilitators made shifts in their decentering as revealed by their attention or lack of attention to a PLC member’s perspectives or knowledge, and whether they decided to act on this knowledge during communication.


My research examines the descriptive powers of the framework of subjective probability, introduced by Psychologists Amos Tversky and Daniel Kahneman, on mathematical misconceptions that arise in areas other than probability.  Tversky and Kahneman referred to subjective probability as a probability estimate of an event, either given by a subject or inferred from her or his behavior, and described specific heuristics and biases associated with probabilistic inferences.  In particular, this report examines undergraduate prospective elementary school teachers' use of prime numbers when asked to simplify a "large" fraction in a clinical interview setting. Participants' approaches to the task are interpreted through the framework of judgment under uncertainty: more specifically, the heuristics of representativeness, availability and adjustment from the anchor, as well as their respective biases. The results suggest that participants' struggles associated with elementary number theory originate in the use of subjective probability.


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.



The purpose of this research is to describe the sociomathematical norm of speaking in meaning as well as describe its emergence in a professional learning community (PLC). Speaking in meaning is used to illustrate the type of mathematical participation expected of the teachers. Speaking in meaning implies that the teachers not only base their responses and suggestions on mathematical concepts but also give conceptual descriptions when giving their explanations. The data for this study is currently being collected from a PLC whose members are secondary mathematics and science teachers.  Initial analysis shows that they are beginning to develop criteria for what is acceptable. For example diagrams used must be clearly labeled and any operations performed must be explained conceptually (not just procedurally). This research has the potential to contribute to the theoretical construct of speaking in meaning as well as inform the instructional design of PLC?s.


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.


When faced with a problem, there are many ways to orientate oneself to it. Are you looking for a specific answer or do you want to understand the conceptual underpinnings of the problem? We will present findings from a study of the science, technology, engineering, and mathematics (STEM) process and dispositional behaviors of high school math and science teachers while involved in a professional development program consisting of four graduate courses aimed at changing teaching practice to more inquiry-based methods. We will argue that one’s orientation to problem solving has a significant impact on learning outcomes and, in the case of teachers, teaching practice.


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.    


A taxonomy for mathematical knowledge-expertise was developed during a year-long study of students' understanding of proof across the undergraduate mathematics major at a medium-size comprehensive university. The taxonomy takes the form of a matrix with elements adapted from science assessment and expertise theory. The matrix was developed in order to more accurately describe the performance of 12 students and one faculty expert on a "proof-aloud" task.  The taxonomy matrix characterizes three stages of mathematical expertise across six cognitive and two affective components of mathematical knowledge. This paper will describe how the taxonomy was developed, what it implies about teaching practice, and its subsequent application to analyzing videotapes of mathematics majors from large research university engaged in problem solving sessions.


The research reported here investigates the question, what is the nature of students’ understanding of first-order differential equations [FODEs] in a modern course on ordinary differential equations [ODEs]? Modern courses on ODEs emphasize analytical, numerical, and qualitative solution methods and hence use graphical and algebraic representations of ODEs and their solution functions. Sfard’s (1991, 1994) theory of reification predicts that a deep understanding of function is a necessary component upon which understanding of FODEs is built; in the terminology of the theory one must have reified the concept of function. The two case studies reported on in this talk show a contrast in understanding that gives insight into the cognitive importance of a reified notion of function to the development of FODE understanding in a modern, multi-representational approach to their study.

Laurie D. Edwards Saint Mary's College of California ledwards@stmarys-ca.edu

The purpose of the research is to investigate the cognitive mechanisms involved in the construction and understanding of proof. The central research question is:  what are the conceptual mechanisms that make it possible for mathematicians, mathematics instructors and students to construct a notion of proof, given their existing knowledge and experience? The data collected will include oral language, written symbols, gesturing, drawing, formal graphing, and other modalities utilized when teaching, talking about, and creating proofs. The primary data source will be videotapes of mathematicians, students and instructors in both interview and naturalistic settings. The analytic framework draws from cognitive linguistics, which utilizes language as well as other modalities to infer the unconscious conceptual mechanisms and source domains that are involved in the construction of new knowledge. Preliminary results from a pilot study indicate differences in language and gesturing when university instructors discuss different kinds of proofs.

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.

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.

Kelly Finn University of Iowa  kelly-f-finn@uiowa.edu

The purpose of this study is to characterize the types of instructional strategies used by college mathematics professors in courses in which perspective secondary (grades 7-12) teachers enroll.  In this study, online surveys are sent to the combined membership list of the AMS and MAA in the United States as a way to gather the data.  The findings from this study emphasize the importance of documenting the instructional strategies of college mathematics professors as a way to determine the extent to which a variety of teaching approaches have been adopted.


Intellectual need, a key part of the DNR theoretical framework, is posited to be necessary for significant learning to occur.  I will introduce the concept of intellectual need and explore ways in which it can be absent from mathematics classrooms.  A case study of two high school algebra teachers whose classes were videotaped illustrates several categories of activity in which students feel little or no intellectual need.  After identifying actions by the teacher that may contribute to this “problem-free activity,” I suggest alternative treatments of some lessons and more general ways to stimulate intellectual need.


As part of a three-semester teaching experiment in calculus, 22 university students collaboratively explored open response tasks. We analyze videodata of students exploring the Quabbin Reservoir Task and presenting their ideas. We study emergence of conceptually important calculus ideas (CICIs) and connections amongst students’ experience, other classes, previous tasks, and CICIs. Based upon our preliminary research, we expect that horizontal and vertical mathematizing, conceptual blending, metonymy, and linguistic invention will be helpful in categorizing ways CICIs and explicit connections are brought forward through students’ explorations. However, these ideas alone are not sufficient to understand the genesis of CICIs or connections. Other important ideas emerging from the analysis are that justification plays a crucial role in the building of CICIs and connections, student questions often focus inquiry and justification, and speculation about what “they” (the professors) want or questioning how current explorations fit with previous class experiences promote connections and formalizing.

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.

Beste Gucler Michigan State University guclerbe@msu.edu

Natasha Speer Michigan State University nmspeer@msu.edu

In this presentation, we share findings from research on knowledge graduate students have about how calculus students think about limits. At the K-12 level, researchers have demonstrated that improving teacher knowledge of student thinking is a powerful model for professional development, leading to changes in teachers’ practices and improvements in student achievement. We hypothesize that this extends to the college level–that improving graduate students’ knowledge of student thinking can improve outcomes for college students. The initial step we take in this larger project is to investigate what graduate students know about their students’ thinking in calculus. We focus in particular on the limit because of the relatively substantial body of research on student thinking and misconceptions in this area. We will present some preliminary findings and also seek audience members’ suggestions for how the methods can be revised to strengthen findings in subsequent studies.

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.

David Hasson San Francisco State University davidhasson@yahoo.com

Eric Hsu San Francisco State University erichsu@math.sfsu.edu

Power differences in mathematical conversations with an authority such as a teacher or  tutor create unintended effects that may alter or interfere with a student's reasoning  during problem solving.  Using ideas from Sfard and Oehrtman, we analyze transcripts of  an authority interviewing second semester calculus students as they attempt to solve a  covariational reasoning problem. We present an analytical framework focused on three  basic areas of student behavior that arise when interacting with a mathematical  authority’s scientific use of language: adaptations to the scientific use of language, probes  of the interviewer for cues of approval regarding these adaptations, and attempts to save  face if adaptations appear unapproved.

Shandy Hauk University of Northern Colorado hauk@unco.edu

Mark K. Davis Learning Helix and Center for Learning and Teaching in the West m.k.davis@comcast.net

Current college mathematics curricula for general education courses in the U.S. lead to a collegiate mathematics education that falls short of meeting national social justice and cultural competence needs. We examine common views of curriculum, explore what it might mean for college level mathematics curricula to be culturally responsive, illustrate the development process for a Liberal Arts Mathematics course, and exemplify what learning from such a curriculum might include. In describing how a culturally responsive curriculum might look, we take the meaning of “curriculum” as a dialogic process associated with situated praxis (Grundy, 1987). We end by outlining some of the research questions that arise around the theory of culturally responsive curricula in college mathematics.

Nate Hisamura Arizona State University nhisamura@cox.net

Arlene Evangelista Arizona State University arlene@mathpost.asu.edu

Michael Oehrtman Arizona State University oehrtman@math.asu.edu

Based on Thompson’s (1994a, 1994b) and Carlson, Jacobs, Coe, Larsen, & Hsu’s (2002) work on covarational reasoning we developed a new covarational framework. The framework consists of three dimensions of conceptual analysis, but this paper will focus on the dimension of mental action. We present a lesson module where teachers are given the task of determining the possible shape of a bottle when they are provided with a graphical representation of the height of water in a bottle as a function of the volume of water. We observed recurring conceptual techniques that were not captured by any of the descriptions of the mental actions in the original framework developed by Carlson et. al (2002). In this paper, we propose new parameterized versions of the previous mental actions and provide a description and examples for one of these new categories in contrast to the previous mental actions described by the original framework.

Matthew Inglis University of Warwick m.j.inglis@warwick.ac.uk

Juan Pablo Mejia-Ramos University of Warwick j.p.mejia@warwick.ac.uk

There is a widespread belief in the mathematics education community that students should be encouraged to avoid basing their level of conviction in mathematical arguments on the authority of the argument’s source. In this presentation we report an experiment which investigated the role that authority plays in the argument evaluation strategies of undergraduate students and research active mathematicians. Our data show that both groups were more persuaded by an argument if it came from an authority figure. The implications of this finding are discussed. It is argued that the role of authority in mathematical argumentation – both in terms of actual behaviour and of normative behaviour –requires deeper scrutiny.


Definitions are one of the most important parts of mathematics. The importance of understanding formal definitions in teaching and learning mathematics has been discussed in the literature. This research explores students? understanding of a new definition and the connections they make from this new definition to the concepts they have previously learned. Eight first year undergraduate students participated in this study. Before the interview, the participants were given a definition that they have never seen before. During the interview, they were presented with a set of tasks, which examined their understanding of the new definition as well as their general understanding of elementary combinatorics. In this report, I will examine the ways that these students attempt to understand this new definition, and how they use this new definition to solve one of the problems that was given to them during the interview.

Jessica Knapp Pima Community College knapp@mathpost.asu.edu or jlknapp@pima.edu

Mathematicians hold a variety of different notions concerning the purposes and necessary or sufficient conditions for proof. Thus it is not surprising that mathematics majors in an advanced calculus course found their professors from different courses each exhibited diverse expectations of proofs produced by the students. The unintended consequence of these varied expectations was that the students developed a notion of an active audience for the proof. Hence a proof was determined acceptable based on for whom it was written. The purpose of this talk is to discuss the development of an active audience in proof writing, its consequences and open the discussion as to which notions of proof it would benefit students to develop.

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.


In this paper, the notion of a record-of / tool-for transition is introduced. This transition involves an important shift in the role played by a form of notation. This transition occurs when a form of notation that has previously been by students primarily as a way to record their informal mathematical activity begins to be used as tool by students to support more formal mathematical reasoning.  Examples from both an abstract algebra context and a differential equation context will be used to illustrate the notion of a record-of / tool-for transition. A theoretical discussion will situate the notion of a record-of / tool-for transition relative to other constructs within the theory of realistic mathematics education (RME).

Christine Larson Indiana University larson.christy@gmail.com	Jill Nelipovich San Diego State University jnelipov@ucsd.edu	Chris Rasmussen San Diego State University chrisraz@sciences.sdsu.edu
Michael Smith San Diego State University msmith25@gmail.com	Michelle Zandieh Arizona State University zandieh@asu.edu

Linear algebra poses a number of significant challenges for students that need to be better understood in order to improve instruction and student understanding.  At the time of this seminar we will have just begun a teaching experiment intended to explore these challenges.  This preliminary report session will be a “working session” in which we bring together participants to examine and discuss the potential for specific modeling tasks to help make the difficult transition to the formalism of linear algebra.  We anticipate that this session will also provide many opportunities for us to have extended discussions with interested participants at other times during the conference.

Joanne Lobato San Diego State University lobato@saturn.sdsu.edu

Issues of transfer typically arise in collegiate math classrooms when students don't perform as well as expected on "application" problems. This paper challenges one's assumptions about transfer by introducing the actor-oriented transfer (AOT) perspective. Three principles arising from AOT research will be presented and instructional implications discussed.

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.

Elise Lockwood Portland State University elockwoo@pdx.edu

Sean Larsen Portland State University slarsen@pdx.edu

Joanna Bartlo Portland State University joannamd@aol.com

The project is designed to contribute to existing research on undergraduate students’ learning and understanding of quotient groups. This report focuses on the first of a series of teaching experiments aimed at developing an instructional approach that supports the reinvention of the quotient group concept. The results of this first iteration of the design-research cycle will be presented, with an emphasis on sharing both preliminary discoveries and new questions that have arisen.

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.

Vilma Mesa University of Michigan vmesa@umich.edu

Preliminary report on an interview study of 14 mathematics faculty in different higher education settings about the ways in which they use mathematics textbooks for instruction. The instructors vary in terms of their teaching experience, their research interests, the type of courses they teach, and whether they themselves have written mathematics textbooks. The study is geared towards understanding the role that textbooks could play in assisting instructors in developing their mathematics teaching expertise.
Activity theory serves as theoretical framework for studying the different activities that surround textbook use by instructors. Preliminary results suggest that instructors' textbook use varies depending on the level of course taught (upper or lower division courses) and by the course content (applied or not). It is less clear that instructors see the textbook as a source for assisting them in improving their teaching, even though they recognize its usefulness for lesson planning.

Richard Millman University of Kentucky millman@ms.uky.edu

Matthew Wells University of Kentucky mwells@ms.uky.edu

This article gives a construct of a year long course sequence of math content for future elementary teachers (pre-service teachers, or PSTs) at a research University.  The sequence is conceptual based and integrates assessment into it. Four issues motivate the courses: (1) the lack of conceptual understanding by the PST, (2) the misunderstanding of the objective of the sequence by the PST, (3) incorporating NCTM Principles and Standards as a central focus, and (4) the need to give PSTs more confidence with mathematics and their ability to teach themselves the mathematics of elementary school. Throughout, we describe how we constructed the course and discuss the importance of the different tools used in the course. With this change in place, we discuss a measure of the effectiveness of this course on PSTs, and conclude with our data analysis. 

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.

Mika Munakata Montclair State University munakatam@mail.montclair.edu

This preliminary research report will present results of an exploratory study on the role of an instructional unit on undergraduate students’ logical reasoning skills.  The instructional unit is based on the pragmatic reasoning schema and is designed to encourage the use of mathematical reasoning on “permission and obligation” problems.  Namely, students will translate traffic and parking regulation signs into logic statements and use these statements to analyze various situations.  Data will be collected from students enrolled in a mathematics course for liberal arts majors and pre-service secondary school teachers enrolled in a college geometry course.  Pre- and post-instructional tests will be administered and students’ responses and ability to transfer skills between context-sensitive and syntactic problems will be analyzed.  The preliminary data will be used to reassess the research methods and refine the instructional unit.  Classroom implications, as well as future di rections for the study, will be discussed. 


In this paper we investigate how kinesthetic experiences can play the role of “bridges” that experientially bring together partial results obtained by symbol manipulation with certain “states of affairs” that students have engaged with physically. The selected interview episodes show that kinesthetic experience can transfer or generalize to the building and interpretation of formal, highly symbolic mathematical expressions. The paper analyzes selected episodes from open-ended individual interviews with three students who had taken, or were taking, a class on differential equations. In the interviews students engaged in a number of different tasks involving a physical tool called the water wheel.


This session will describe the design research on a long-term interdisciplinary professional development program for secondary science and mathematics teachers. The intervention involves intensive work with university faculty and school leaders in mathematics, physics, chemistry, biology, geology, and engineering to develop a four course graduate course sequence. A central challenge in this work is to maintain coherence in the implementation, evaluation, and research. We aim to achieve this coherence through a focus on unifying reasoning patterns, unifying process behaviors, and unifying dispositional behaviors across the project disciplines. In this research report, we provide an overview of the development of our theoretical frameworks in two of these three areas, focusing on covariational reasoning as a unifying reasoning pattern and establishing an inquiry orientation as a unifying process behavior. We illustrate through use of video data how our frameworks and interventions have been adapted over three iterations of the intervention.


Researchers hoped to show that students in an inquiry-oriented, application-based course dealing with college algebra topics could gain in their conceptual underpinnings without sacrificing mechanical skills.  Data from the study confirm one hypothesis and refute the other.  Specifically, the study investigated the achievement of two clusters of students on two measures of linear and exponential reasoning.  Students in a traditional College Algebra course outperformed students in a “reformed” Algebraic Modeling course on a standardized instrument emphasizing procedural and mechanical skills.  Students in the Modeling course had significantly higher scores on the instrument developed for the study, that emphasized conceptual understandings.  One conclusion drawn by the study is that a modified curriculum and pedagogy can produce alternate outcomes, while leaving open the question of what types of outcomes are most desirable.


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.


The research builds on the Inquiry-Oriented Differential Equations (IO-DE) project to develop a model for how it is that teachers create and sustain inquiry-oriented mathematics classrooms to support students’ learning mathematics in powerful and deep ways. For the purpose, we have conducted a semester long classroom teaching experiment. The analysis of this research focuses on the teacher’s questioning in the context of students’ argumentation in an IO-DE course. The analysis identifies five types of questions used by the teacher in the IO-DE: Evaluative questions, Requests to explain thinking, Requests to justify thinking, Requests to check or assess student progress, and Clarifying questions. This research describes the roles of these questions in the context of students’ argumentation to reveal how the teacher strategically applies questions for the construction of mathematics. This research implies that teachers’ knowledge of questioning is of essence for the improvement of mathematics instruction. 


We discuss the types of justifications offered by two high-school algebra teachers, based on classroom observations during a two-year period. These are classified using the proof scheme taxonomy of Harel and Sowder and illustrated with examples. Justifications based on authority or on empirical evidence predominate over deductive ones. The type and quality of justifications offered can be limited by teachers’ pedagogical knowledge as well as their mathematical (content) knowledge.


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.  During class, students watched and discussed videos of “competent” problem solvers talking aloud while proving several statements that are normally difficult for students at this level.  The goal of the session was to help students connect informal and formal aspects of proof.  During the following class, students were interviewed in both small group settings and a whole class setting about their reaction to watching the video.  This talk will center on the relative advantages and disadvantages of this type of intervention, and will solicit ideas for improvement for the third iteration.


This preliminary report focuses on an effort to understand how students make sense of mathematical statements involving multiple quantifiers and to help students learn how to do so in a way that is consistent with mathematical convention. Results are presented from a collection of mini teaching experiments conducted with the aim of developing an instructional approach that builds on students’ own mathematical activity.


This is a preliminary report on a study comparing student achievement between traditional and distance learning versions of the same college algebra course.  The traditional cohort is a group of 37 college freshmen from the Appalachian area while the distant cohort is 38 rural, primarily Appalachian high school seniors. We will discuss the efforts to provide a comparable content and instructional experience to both groups. All students are completing the same coursework. Examinations are uniformly graded. The local tutoring support for traditional students is matched by real-time e-tutoring for the distant cohort. In addition to achievement, the study is designed to identify critical elements of an effective model for a distance learning college algebra course. We use an exploratory investigation with a concurrent design, involving examining student scores and survey data. The mixed methods used in this study allowed for additional insights into the attitude of distance learning in.


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 will report on results from a teaching experiment that used 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 the structure of the Riemann integral with college calculus students.  Students were able to successfully assimilate the Riemann integral structure into an already established limit structure (via approximations).  Their struggles were mainly concentrated in areas where the particulars of Riemann sums departed from the limit structures students had developed while learning about limits of functions, limits of sequences, and the definition of the derivative.


This study explores the ways in which eleven preservice elementary teachers used a web-based teacher resource to apply a mathematical definition, to correct a procedural error in arithmetic, and to make sense of a story requiring the multiplication of fractions. In our analysis we propose a framework to compare the behaviors and values expressed by our participants with the values and norms of the mathematical community. This analysis suggests that many preservice elementary teachers are profoundly mathematically unsophisticated. In other words, they displayed a set of values and avenues for doing mathematics so different from that of the mathematical community, and so impoverished, that they found it difficult to create fundamental mathematical understandings.
 

In this presentation, we summarize our search for empirical research on collegiate mathematics teaching practices. Where research about teaching is relatively common, descriptive empirical research on what collegiate mathematics teachers actually do as they teach is virtually non-existent. Our claim is based on a review of peer-reviewed journals where research on collegiate mathematics teaching is published. Because such research is needed (to illustrate innovative practices, promote more research on practice, and support beginning teachers? learning about teaching) we also propose a framework for studying collegiate teaching practice. Its development was informed by a similar frame for K-12 teaching (NCTM, 1991) and our analysis of teachers’ actions and decision-making in the most common instructional format, lecture presentation. We assert that college mathematics teachers make many important decisions as they plan and carry out their instruction and much can be learned by examining those practices.


This research is part of a larger investigation designed to explore what it means for students to understand the Calculus. The initial phase of the study seeks to comprehensively examine the  perceptions of well-known experts in the fields of mathematics to identify the concepts and skills important to the calculus curriculum and delineate a number of mathematical problems these same experts believe could be used to assess students' level of understanding of those concepts and skills. Thirty  participants will be identified and interviewed to collect information related to their knowledge and beliefs about student understanding, the calculus curriculum, and assessment. Interview data will be transcribed and analyzed using methods of categorical content analysis to extract themes and patterns. Initial findings from the analysis of the interview data will be reported. Discussions of the research questions, design, and preliminary findings will help to refine the future direction of the study.


Weber (2001) describes four types of strategic knowledge (Hart, 1994) that assist students with proof-writing. They are knowledge of: (1) proof techniques, (2) important theorems, (3) usefulness of theorems, and (4) timing for syntactic strategies. In comparing doctoral and undergraduate students’ abstract algebra proof constructions, Weber discovers that such strategic knowledge not found in undergraduates is possessed by graduate students. In this report we describe somewhat conflicting evidence to Weber’s results. Our findings indicate that although our undergraduate students enrolled in abstract algebra possess proof techniques, know the important theorems, and are able to see the usefulness of a theorem, they continue to struggle with proof-writing. One contributing obstacle to this phenomena is symbol manipulation, which Weber briefly discusses. We define the second obstacle as interfering knowledge; this is knowledge possessed by a student that conflicts with the new content or interferes with the ability to carry out a proof.


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 paper discusses whether the level of inquiry orientation in teachers’ Professional Learning Community (PLC) provides a means to characterize the group’s effectiveness. The PLCs in this study consist of high school mathematics and science teachers participating in a professional development project. By focusing on the interactions between the teachers, we hypothesized that the inquiry orientation (the group’s inclination to engage in problematic issues, to address common problematic issues, and to work to bring resolution to the problematic issues) could be a good predictor of effectiveness. We use this perspective to contrast two PLC groups: one that was considered to be very ineffective and one that was considered to have productive discourse. The inquiry orientation of the members of the PLC seems to have a strong influence on the discourse within the group.


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.


As computers become a part of everyday life, many students are receiving instruction in online environments. Universities, corporations, and even hobbyists are producing online course materials made available for general public use. What differentiates these courses, and what features of an online instructional presentation contribute to its effectiveness? In this report, I address these questions by proposing and applying an evaluative framework to five online introductory calculus courses. The framework is informed by cognitive research and research in mathematics education. As a demonstration of the framework, I analyzed each course’s instruction for the chain rule, a procedure that is mathematically interesting, commonly encountered within and outside of calculus, and challenging for the student. The application of this framework revealed that the courses represented very different ideas on what it means to ``know'' calculus. In particular, the roles of rigor, exploration, computation, and intuition in learning received varying amounts of attention.


The concept of function is first presented to students in the setting of real numbers and the Cartesian plane. The broad definition of function, as a relation where to every input corresponds only one output, is complemented by representations specific to the rectangular coordinate system, like the vertical line test. When students extend their work with functions to other contexts, such as the polar coordinate system, these particular representations often replace the universal definition of function.  Students rejected   as a function in polar coordinates because, as a circle, it “doesn’t pass the vertical line test”. This study is a prelude to further work on the concept of function in the multivariate context, and was designed to measure how students deal with generalizations of the function concept outside of the Cartesian coordinate setting, but still in the single variable context.


Orchestrating class discussions of mathematics in ways that conform to reform-based principles of mathematics instruction can be challenging to instructors accustomed to lecture-oriented classrooms. In particular, instructors are challenged to direct classroom activities and discussion in mathematically productive ways, while simultaneously encouraging students to seek mathematical authority not in their teacher, but in their own mathematical reasoning and judgment. In this report, I investigate the experiences of two university mathematicians during their first attempts to foster whole-class dialogue and argumentation using an inquiry-oriented differential equations curriculum. I argue that differences in class outcomes were associated with the different ways each instructor negotiated the tension between serving as a mathematical authority and enabling students to find mathematical authority in legitimate mathematical reasoning. Balancing these roles shapes opportunities for student learning, and understanding such effects offers support to university mathematics instructors who wish to adopt reform-based principles of instruction in their classrooms.


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.


There are calls for greater communication between mathematics departments and the various client departments whose students require a strong foundation in mathematics from organizations including the MAA and Project Kaleidoscope (PKAL). Buffalo State, as part of a U.S. Department of Education Title III grant, has formed a working group comprised of mathematics, science, technology, and computer information systems faculty whose objective is to gain understanding of the needs of the various client departments, assess students’ abilities in identified areas, and share the findings with departments. The discussion of results will allow for building better understanding among departments and generating ideas for the improvement of instruction and student learning to be delivered through a series of faculty workshops. The focus of this report is on the development and administration of a draft assessment instrument. Analysis of initial pilot results will be discussed along with implications for undergraduate mathematics courses serving students in our partner disciplines.


This report describes an empirical study of the lesson planning practices of seven novice graduate student instructors (GSIs) teaching a university precalculus course.  We describe the salient features of the processes that the GSIs typically followed when planning their lessons, the amount of time spent on different parts of this process, and the resources that the GSIs employed while crafting their lessons.  The GSIs in this study typically formulated objectives, consulted resources, selected activities and (in some cases) attempted to manage instructional time while planning.  We found that GSIs typically devoted less and less time to lesson planning during their first semester of teaching, and that they typically use the resources that were provided to them (prepared lesson plans and the course textbook), although these were used a starting point for developing more personalized lessons, rather than as exact “blueprints” for instruction.


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.


A partnership-driven approach to the creation, implementation and assessment of a set of high quality mathematics courses in pre-service teacher education is described within the framework of a comprehensive, rural Mathematics and Science Partnership (MSP).  The MSP course development teams are mathematics disciplinary and education faculty at nine Institutes of Higher Education (IHEs) and K-12 mathematics teacher from 51 school districts in the central Appalachian regions of Kentucky, Tennessee, and Virginia. Evaluations of sixteen mathematics courses, the enrolled pre-service teachers and the teacher education programs will be discussed relative to their impact on teacher education programs in the participating IHEs.  These include a longitudinal study of the demographic and academic characteristics of the pre-service teachers and the course impact on their cognitive, affective and pedagogical domains.  The sustained nature of the partnership approach to improvement of content and teacher quality will be highlighted in the presentation.


The purpose of this report is to describe how several seemingly different phenomenon in mathematics education can be seen through one unifying lens with the use of Fauconnier and Turner’s theory of conceptual blending. This theory describes how humans reason and learn by combining familiar mental spaces or frames into a new blended space or an integrated network of blended spaces. I use three examples of theoretical frameworks from the mathematics education literature, including data analyzed within these frameworks, and show how they may be reframed using conceptual blending.  The examples are 1) the notion of the key idea of a proof, 2) metaphorical and metonymic relationships in a structured understanding of the concept of derivative, 3) the emergent models heuristic from realistic mathematics education. By explicating these three examples I hope to illustrate how conceptual blending may play a role in our work in mathematics education and to engage our community in a discussion of these possibilities.