CRUME 2008 Proceedings

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

Conference on Research in
Undergraduate Mathematics Education

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

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

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

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

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

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

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

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

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

2008 BEST PAPER AWARD

Exploring Epistemological Obstacles to the Development of Mathematical Induction

Stacy Brown
Pitzer College
stacy_brown@pitzer.edu

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

In September 2007, 150 mathematics educators from 92 institutions met in Kansas City for the conference, Doctoral programs in mathematics education: Progress in the last decade. The goals of the conference were to exchange information and develop plans for improving the quality of mathematics education doctoral programs. In this panel session, we will facilitate a discussion on issues from the conference relevant to the RUME community. In particular, we will discuss challenges that RUME doctoral programs face and raise questions that might be useful topics for future investigation. For example, one finding from the research leading up to the conference was that the majority of mathematics education doctorates do not take university jobs (e.g., some take jobs in K-12 teaching and administration, in governmental agencies, or educational assessment and publishing). Open questions to consider: What do RUME doctorates do? What percentages take research positions in mathematics departments? …in education departments? …teach at community colleges? …do other work? And, what consequences does this have for how we design programs to prepare students for their futures?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

2008 BEST PAPER AWARD: Honorable Mention

An Investigation of Graduate Teaching Assistants’ Statistical Knowledge for Teaching

Jennifer Noll
Portland State University
noll@pdx.edu

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

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

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

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

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

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

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

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

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

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

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

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

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

The purpose of the ongoing research is to generate insights into how students may come to understand the formal definition of limit of a function at a point, and to move toward the elaboration of a cognitive model of what might be entailed in coming to understand this formal definition. Specifically, we aim to: 1) develop insight into students’ reasoning in relation to their engagement in principled instruction designed to support their reinventing the formal definition of limit at a point; and, 2) inform the design of instruction that might support students in reinventing the formal definition of limit.

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

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

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

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

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

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

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

2008 BEST PAPER AWARD: Honorable Mention

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

Keith Weber
Rutgers University
keith.weber@gse.rutgers.edu

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

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

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

Aditya P. Adiredja, University of California Berkeley aditya@berkeley.edu	Randi A. Engle University of California Berkeley raengle@berkeley.edu	Danielle Champney University of California Berkeley ddchamp@gmail.com	Amy Huang University of California Berkeley aihuang@ucdavis.edu
Mark Howison University of California Berkeley mhowison@berkeley.edu	Niral Shah University of California Berkeley niral@berkeley.edu	Pegah Ghaneian, University of California Berkeley pghaneian@berkeley.edu

Todd A. Grundmeier Cal Poly - San Luis Obispo tgrundme@calpoly.edu	Jacey Branchetti Cal Poly - San Luis Obispo jabranch@calpoly.edu
Joaquin Castillo Cal Poly - San Luis Obispo ljcastil@calpoly.edu	Carla Scherer Cal Poly - San Luis Obispo cmschere@calpoly.edu

Shandy Hauk University of Northern Colorado hauk@unco.edu	Hollylynne Lee North Carolina State University hollylynne@ncsu.edu
Karen Marrongelle Portland State University karenmarpdx.edu	Keith Weber Rutgers University keith.weber@gse.rutgers.edu

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