Conference Program/Abstracts

Abstracts - Contributed Reports

Switcher and Persister Experiences in Calculus I

Jessica Ellis, Chris Rasmussen, and Kristin Duncan

Abstract: Previous reports show that not only are too few students pursuing Science, Technology, Engineering, or Mathematics (STEM) fields, but also many who originally intend to pursue these fields leave after their experiences in introductory STEM courses. Based on data gathered in a national survey, we will present an analysis of 5381 STEM intending students enrolled in introductory Calculus in Fall 2010, 12.5% of whom switched out of a STEM trajectory after their experience in Calculus I. When asked why these students no longer intended to continue taking Calculus (an indicator of continuing their pursuit of a STEM major), 31.4% cited their negative experience in Calculus I as a contributing factor. We analyze student and their instructor survey responses on various aspects of their classroom experience in Calculus I to better understand what aspects of this experience contributed to their persistence.

Critiquing the Reasoning of Others:
Devil’s Advocate and Peer Interpretations as Instructional Interventions

Aviva Halani, Owen Davis, and Kyeong Hah Roh

Abstract: This study investigated the ways in which college mathematics teachers might encourage the development of student reasoning through critiquing activities. In particular, we focused on identifying situations in which the instructional interventions were implemented to encourage the critiquing of arguments and in which students explained another’s reasoning. Data for the study come from two teaching experiments – one from the domain of combinatorics and the other from real analysis. Through open coding of the data, Devil’s Advocate and Peer Interpretations emerged as effective interventions for the creation of sources of perturbation for the students and for assisting in the resolution of a state of disequilibrium. These two interventions differ in design and in the type of reasoning students evaluate, but they both provoke students to further develop their reasoning, and therefore their understanding. We discuss the implications of these interventions for both research and teaching practice.

Entity versus Process Conceptions of Error Bounds in Students’ Reinvention of Limit Definitions

Robert Raish, Michael Oehrtman, Jason Martin, Brian Fisher, and Craig Swinyard

Abstract: We report results from a guided reinvention of the definition of sequence convergence conducted in three second-semester calculus classes. This report contributes to the growing body of research on how students come to understand and reason with formal limit definitions, focusing on the emergence of students’ understanding of the epsilon quantity, conceived in terms of error bounds. Using Sfard's framework of the condensation of processes to entities, we mapped the possible conceptual trajectories followed by the students in the study. In this report, we detail our map, these trajectories and students’ reasoning about other aspects of the formal definition, and the influence of reasoning about approximations and error analyses in students’ progression.

Under the Radar: Foundational Meanings that Secondary Mathematics Teachers Need, Do Not Have, but Colleges Assume

Pat Thompson, Neil Hatfield, Cameron Byerly and Marilyn Carlson

Abstract: High school mathematics teachers must have coherent systems of mathematical meanings to teach mathematical ideas well. One hundred five teachers were given a battery of items to discern meanings they held in with respect to quantities, variables, functions, and structure. This paper reports findings on a sample of items that, by themselves, should alert college mathematics professors that foundational understandings they assume students have in advanced mathematics courses likely are commonly missing.

Using Cognitive Science with Active Learning in a Large Lecture College Algebra Course

David Miller and Matthew Schraeder

Abstract: At a research university near the east coast, researchers have restructured a College Algebra course by formatting the course into two large lectures a week, an active recitation size laboratory class once a week, and an extra day devoted to active group work called Supplemental Practice (SP). SP was added as an extra day of class where the SP leader has students work in groups on a worksheet of examples and problems, based off of worked-example research, that were covered in the previous week’s class material. Two sections of the course were randomly chosen to be the experimental group and the other section was the control group. The experimental group was given the SP worksheets and the control group was given a question-and-answer session. The experimental group significantly outperformed the control on a variety of components in the course, particularly when the number of SP days was analyzed.

On the Plus Side: A Cognitive Model of Summation Notation

Steve Strand and Sean Larsen

Abstract: This paper provides a framework for analyzing and explaining successes and failures when working with summation notation. Cognitively, the task of interpreting a given summation-notation expression differs significantly from the task of expressing a long-hand sum using summation notation. As such, we offer separate cognitive models that 1) outline the mental steps necessary to carry out each of these types of tasks and 2) provide a framework for explaining why certain types of errors are made.

Preservice Elementary Teachers’ Understanding of Greatest Common Factor Story Problems

Kristen Noblet

Abstract: Little is known about preservice elementary teachers’ mathematical knowledge for teaching number theory concepts, like greatest common factor or GCF. As part of a larger case study investigating preservice elementary teachers’ understanding of topics in number theory, both content knowledge and pedagogical content knowledge (Shulman, 1986), a theoretical model for how preservice elementary teachers understand GCF story problems was developed. An emergent perspective (Cobb & Yackel, 1996) was used to collect and analyze data in the form of field notes, student coursework, and responses to task-based one-on-one interviews. The model resulted from six participants’ responses to three sets of interview tasks where participants discussed concrete, visual, and story problem representations of GCF. In addition to discussing the model and relevant empirical evidence, I suggest language with which to discuss GCF representations.

Students’ Conceptions of Mathematics as a Discipline

George Kuster

Abstract: Researchers have found that students’ beliefs about mathematics impact the way in which they learn and approach mathematics in general. The purpose of this study is to categorize college students’ various conceptions concerning mathematics as a discipline. Results from this study were used to create a preliminary framework for categorizing student conceptions. The results of this study indicate that the conceptions are numerous and range greatly in complexity. The results also suggest the need for further study to qualify the various student conceptions and the roles they play in students' understanding of and approach to performing mathematics.

Using Disciplinary Practices to Organize Instruction of Mathematics Courses for Prospective Teachers

Yvonne Lai

Abstract: One challenge of teaching content courses for prospective teachers is organizing instruction in ways that represent the discipline with integrity while serving the needs of future teachers—for example, choosing math problems that provide a logical development of a topic while also addressing mathematical knowledge for teaching. This paper examines the work entailed in structuring in-class work in mathematics courses for teachers. It argues that practices of teaching that are mathematical—such as representing ideas, grounding reasoning in mathematical observations available to the class, using definitions, or using mathematical language—can be used to negotiate mathematical and pedagogical aims, and therefore can be used to organize instruction of mathematical knowledge for teaching while simultaneously developing a disciplinary understanding.

On the Emergence of Mathematical Objects:
The Case of e^(az)

Ricardo Nemirovsky and Hortensia Soto-Johnson

Abstract: In this report we propose an alternate account of mathematical reification as compared to Sfard’s (1991) description, which is characterized as an “instantaneous quantum leap”, a mental process, and a static structure. Our perspective is based on two in-service teachers’ exploration of the function , using Geometer’s Sketchpad. Using microethnographic analysis techniques we found that the long road to beginning to reify the function entailed interplay between body-generated motion and object self-motion, kinesthetic continuity between different sides of the “same” thing, cultural and emotional background of life with things-to-be, and categorical intuitions. Our results suggest that perceptuomotor activities involving technology may serve as an instrument in facilitating reification of abstract mathematical objects such as complex-valued functions.

Commonly Identified Students’ Misconceptions About Vectors and Vector Operations

Aina Appova and Tetyana Berezovski

Abstract: In this report we present the commonly identified error patterns and students’ misconceptions about vectors, vector operations, orthogonality, and linear combinations. Twenty three freshmen students participated in this study. The participants were non-mathematics majors pursuing liberal arts degrees. The main research question was: What misconceptions about vector algebra were still prevalent after the students completed a freshmen-level linear algebra course? We used qualitative data in the form of artifacts and students’ work samples to identify, classify, and describe students’ mathematical errors. Seventy four percent of students in this study were unable to correctly solve a task involving vectors and vector operations. Two types of errors were commonly identified across the sample: a lack of students’ understanding about vector operations and projections, and a lack of understanding (or distinction) between vectors and scalars. Final results and conclusions include research suggestions and practitioner-based implications for teaching linear algebra in high school and college.

Preparing Students for Calculus

April Brown Judd and Terry Crites

Abstract: This quantitative study compared the implementation of a problem-based curriculum in precalculus and a modular-style implementation of traditional curriculum in precalculus to the historical instructional methods at a western Tier 2 public university. The goal of the study was to determine if either alternative approach improved student performance in precalculus, improved student efficacy around learning mathematics and better prepared students for success in a calculus sequence. The study used quantitative data collection and analysis. Results indicate students who experienced the problem-based curriculum should be better prepared to learn calculus but mixed results in terms of retention and success in calculus.

Partial Unpacking and Indirect Proofs: A Study of Students' Productive Use of the Symbolic Proof Scheme

Stacy Brown

Abstract: This paper examines mathematics majors' evaluations of indirect proofs and of the compound statements used in such forms of proofs. Responses to survey items with a cohort of 23 students and six 1-hour clinical interviews, indicate that the students who could successfully evaluate indirect arguments and who could successfully recognize logically equivalent statements, tended to use partially unpacked (Selden & Selden, 1995) versions of the statement and the proofs and, in so doing, demonstrated a productive use of the symbolic proof scheme. Whereas, both successful and unsuccessful students tended to use proof frameworks (Selden & Selden, 1995). Moreover, successful students' approaches are suggestive of activities, which are rarely found in introductory proof texts, yet may benefit novice proof writers.

Utilizing Types of Mathematical Activities to Facilitate Characterizing Student Understanding of Span and Linear Independence

Megan Wawro and David Plaxco

Abstract: The purpose of this study is to investigate students’ concept images of span and linear (in)dependence and to utilize the mathematical activities of defining, example generating, problem solving, proving, and relating to provide insight into these concept images. The data under consideration are portions of individual interviews with linear algebra students. Grounded analysis revealed a wide range of student conceptions about the span and/or linear (in)dependence. The authors organized these conceptions into four categories: travel, geometric, vector algebraic, and matrix algebraic. To further illuminate participants’ conceptions of span and linear (in)dependence, the authors developed a framework to classify the participants’ engagement into five types of mathematical activity: defining, proving, relating, example generation, and problem solving. This framework could prove useful as a means of providing finer-grained analyses of students’ conceptions and the potential value and/or limitations of such conceptions in certain contexts.

Coherence from Calculus to Differential Equations

Jennifer Czocher, Jenna Tague, and Greg Baker

Abstract: Despite recent research efforts to make calculus more coherent with other fields, instructors still express dissatisfaction in the mathematical preparation of their students. Even further, we suggest that there are coherence issues within the field of mathematics. In this paper, we expose and examine an epistemological mismatch between how calculus is expected to be known in calculus and how calculus is expected to be used in differential equations.

Odd Dialogues on Odd and Even Functions

Dov Zazkis

Abstract: A group of prospective mathematics teachers was asked to imagine a conversation with a student centered on a particular proof regarding the derivative of even functions and produce a script of this imagined dialogue. These scripts provided insights into the script-writers’ mathematical knowledge, as well as insights into what they perceive as potential difficulties for their students, and by extension difficulties they may have had themselves when learning the concepts. The paper focuses on the script-writers’ understandings of derivative and of even/odd functions.

Secondary Teachers’ Development of Quantitative Reasoning

David Glassmeyer, Michael Oehrtman, and Jodie Novak

Abstract: This study was designed to document the development of teachers’ ways of thinking about quantitative reasoning, one of the standards for mathematical practice in the Common Core State Standards. Using a models and modeling perspective, the authors designed a model-eliciting activity (MEA) that was implemented in a graduate mathematics education course focusing on quantitative reasoning. Teachers were asked to create a quantitative reasoning task for their students, which they subsequently revised three times in the course after receiving instructor, peer, and student feedback. The MEA documented the development of the teachers’ models of quantitative reasoning, and the findings of this study detail one group of three teachers’ development over the course. Findings include an overall model of teachers’ development that is both generalizable and sharable for other researchers and teacher educators.

Students Reconciling Notions of One-to-One Across Two Contexts

Michelle Zandieh, Jessica Ellis, and Chris Rasmussen

Abstract: This research is part of a larger study. In previous work we created a framework for analyzing student understanding that incorporates five clusters of metaphorical expressions as well as properties and computations that students spoke about when discussing function or linear transformation. In this paper we apply this framework to the setting of students reconciling their understandings of one-to-one in the context of precalculus-type functions with their understandings of one-to-one in the context of linear algebra. Ideally we would like students to be able to recognize a similar structure for one-to-one in each context, and thereby to strengthen their overall understanding of the notion of one-to-one. This proposal provides four vignettes that we found illustrative of the way students reasoned about one-to-one within and across the two contexts. More broadly we find the case of one-to-one as prototypical of the struggles students have in seeing similarities across contexts.

On Mathematics Majors’ Success and Failure at Transforming Informal Arguments into Formal Proofs

Bo Zhen, Juan Pablo Meija-Ramos, and Keith Weber

Abstract: In this paper, we examine 26 instances in which mathematics majors attempted to write a proof based on an informal explanation. In each of these instances, we represent students’ informal explanations using Toulmin’s (1958) scheme, we use Stylianides’ (2007) conception of proof to identify what one would need to accomplish to transform the informal explanation into a proof. We then compare this to the actions that the participant took in attempting to make this transformation. The results of our study are categories of actions that led students to successfully construct valid proofs and actions that may have hindered proof construction.

Mathematicians’ Example-Related Activity When Proving Conjectures

Elise Lockwood, Amy B. Ellis and Eric Knuth

Abstract: Examples play a critical role in mathematical practice, particularly in the exploration of conjectures and in the subsequent development of proofs. Although proof has been an object of extensive study, the role that examples play in the process of exploring and proving conjectures has not received the same attention. In this paper, results are presented from interviews conducted with six mathematicians. In these interviews, the mathematicians explored and attempted to prove several mathematical conjectures and also reflected on their use of examples in their own mathematical practice. Their responses served to refine a framework for example-related activity and shed light on the ways that examples arise in mathematicians’ work. Illustrative excerpts from the interviews are shared, and five themes that emerged from the interviews are presented. Educational implications of the results are also discussed.

Transfer of Critical Thinking Disposition from Mathematics to Statistics

Hyung Kim and Tim Fukawa-Connelly

Abstract: In this study we draw on the constructs of eagerness, flexibility and willingness to characterize the necessary disposition for critical thinking that is required in learning statistics in addition to specific content knowledge (Enis, 1989). We investigated the challenges that students who are highly successful in mathematics might have in doing statistics and found that while a student might have an inquisitive disposition and good proficiency with the foundational mathematical concepts such as functions and function transformations, that same student might struggle in statistics. Even concepts that are seemingly related to their mathematical counterparts such as what is a variable when considering a population and sample may cause problems as the question is distinct enough from the mathematical sense. We suggest that such students may experience greater than usual affective problems in a statistics class and may, therefore, give up easier and earlier than students who were less successful mathematically.

The Emergence of Algebraic Structure: Students Come to Understand Zero-Divisors

John Paul Cook

Abstract: Little is known about how students learn the basic ideas of ring theory. While the literature addressing student learning of group theory is certainly relevant, the concept of zero-divisor in particular is one for which group theory has no analog. In order to better understand how students come to understand zero-divisors, this talk will present results from a study that investigated how students can capitalize on their intuitive notions of solving equations to reinvent the definitions of ring, integral domain, and field. In particular, the emergence and progressive formalization of the concept of zero-divisor at various stages of the reinvention process will be detailed and discussed.

Performance and Persistence Among Undergraduate Mathematics Majors

Joe Champion and Ann Wheeler

Abstract: There is little mixed methods research into the patterns of course taking, performance, and persistence among mathematics majors, in general, and among secondary mathematics majors, in particular. Drawing from a sample of 42,825 mathematics enrollment records at two universities over a six-year period, this study presents quantitative summaries of mathematics majors' performance and persistence in undergraduate mathematics courses alongside qualitative themes from interviews of nine secondary mathematics majors at one of the universities. Implications include potential strategies for mathematics programs and faculty to support the success of mathematics majors in undergraduate mathematics coursework, with special emphasis on prospective secondary mathematics teachers.

Venn Diagrams as Visual Representations of Additive and Multiplicative Reasoning in Counting Problems

Aviva Halani

Abstract: This case study explored how a student could use Venn diagrams to explain his reasoning while solving counting problems. An undergraduate with no formal experience with combinatorics participated in nine teaching sessions during which he was encouraged to explain his reasoning using visual representations. Open coding was used to identify the representations he used and the ways of thinking in which he engaged. Venn diagrams were introduced as part of an alternate solution written by a prior student. Following this introduction, the student in this study often chose to use Venn diagrams to explain his reasoning and stated that he was envisioning them. They were a powerful model for him as they helped him visualize the sets of elements he was counting and to recognize over counting. Though they were originally introduced to express additive reasoning, he also used them to represent his multiplicative reasoning.

In-service Secondary Teachers’ Conceptualization of Complex Numbers

Stephenie Anderson Dyben, Hortensia Soto-Johnson and Gulden Karakok

Abstract: This study explores in-service high school mathematics teachers’ conception of various forms of a complex number and the ways that they transition between different representations’ (algebraic and geometric) of these forms. Data were collected from three high school mathematics teachers via a ninety- minute interview after they completed professional development on complex numbers. Results indicate that these teachers do not necessarily objectify exponential form of complex numbers and only conceptualized it at the operational level. On the other hand, two teachers were very comfortable with Cartesian form and showed process/object duality by translating between different representations of this form. It appeared that our participants’ ability to develop a dual conception of complex numbers was bound by their conceptualization of the various forms, which in turn was hindered by their representations of each form.

The Merits of Collaboration Between Mathematicians and Mathematics Educators on the Design and Implementation of An Undergraduate Course on Mathematical Proof and Proving

Orit Zaslavsky, Pooneh Sabouri, and Michael Thoms

Abstract: The goal of our study was to characterize the processes and to identify the ways in which different kinds of expertise (mathematics vs. mathematics education) unfolded in the planning and teaching of an undergraduate course on Mathematical Proof and Proving (MPP), which was co-taught by a professor of mathematics and a professor of mathematics education. The content of the course consisted of topics that were supposed to be familiar to the students, i.e., high school level algebra, geometry, and basic number theory. In particular, we looked for instances that would help understand how each expertise contributed to the course and complemented the other. The findings indicate that by co-teaching and constantly reflecting on their thinking and teaching, the instructors became aware of the added value of working together and the unique contribution each one had.

Teaching Undergraduate Calculus for Transfer:
A Qualitative Case Study of the Calculus Sequence at One Liberal Arts College

Noelle Conforti Preszler

Abstract: At small liberal arts colleges, a single calculus sequence must successfully accommodate students from various majors, such as mathematics, biology, chemistry, and economics. This qualitative case study considers mathematics professors' perspectives about the required nature of calculus in various disciplines, attempts to identify how calculus instructors teach with the aim of preparing students to apply calculus knowledge in their future coursework, and how the disciplinary focus of their students affects professors' design and teaching of calculus courses. Framed using aspects of teaching and learning shown to promote transfer of knowledge, results suggest that the professors teach for understanding and allow in-class processing time, but could improve their emphasis on applying calculus in non-mathematics disciplines. This study contributes to the growing body of undergraduate mathematics education research intended to document undergraduate teaching practices.

Students’ Axiomatizing in a Classroom Setting

Mark Yannotta

Abstract: The purpose of this paper is to examine descriptive axiomatizing as a classroom mathematical activity. More specifically, if given the opportunity, how do students select axioms and how might their intellectual needs influence these decisions? These two case studies of axiomatizing address these questions and elaborate on how students engage in this practice within a classroom setting. The results of this research suggest that while students may at first be resistant to axiomatizing, this mathematical activity also affords them opportunities to create meaning for new mathematical content and for the axiomatic method itself.

Understanding Abstract Algebra Concepts

Anna Titova

Abstract: This study discusses various theoretical perspectives on abstract concept formation. Students’ reasoning about abstract objects is described based on proposition that abstraction is a shift from abstract to concrete. Existing literature suggested a theoretical framework for the study. The framework describes process of abstraction through its elements: assembling, theoretical generalization into abstract entity, and articulation. The elements of the theoretical framework are identified from students’ interpretations of and manipulations with elementary abstract algebra concepts including the concepts of binary operation, identity and inverse element, group, subgroup. To accomplish this, students participating in the abstract algebra class were observed during one semester. Analysis of interviews and written artifacts revealed different aspects of students’ reasoning about abstract objects. Discussion of the analysis allowed formulating characteristics of processes of abstraction and generalization. The study offers theoretical assumptions on students reasoning about abstract objects. The assumptions, therefore, provide implications for instructions and future research.

Covariational Reasoning and Graphing in Polar Coordinates

Kevin Moore, Teo Paoletti, Jackie Gammaro, and Stacy Musgrave

Abstract: An extensive body of research exists on students’ function concept in the context of graphing in the Cartesian coordinate system (CCS). In contrast, research on student thinking in the context of the polar coordinate system (PCS) is sparse. In this report, we discuss the findings of a teaching experiment that sought to characterize two undergraduate students’ thinking when graphing in the PCS. As the study progressed, the students’ capacity to engage in covariational reasoning emerged as critical for their ability to graph relationships in the PCS. Additionally, such reasoning enabled the students to understand graphs in the CCS and PCS as representative of the same relationship despite differences in appearance. Collectively, our findings illustrate the importance of covariational reasoning for conceiving graphs as relationships between quantities’ values and that graphing in the PCS might create one opportunity to promote such reasoning when combined with graphing in the CCS.

Using Metaphors to Support Students’ Ability to Reason about Logic

Paul Christian Dawkins and Kyeong Hah Roh

Abstract: In this paper, we describe an inquiry-oriented method of using metaphors to support students’ development of conventional logical reasoning in advanced mathematics. Our model of instruction was developed to describe commonalities observed in the practice of two inquiry-oriented real analysis instructors. We present the model via a general thought experiment and one representative case study of a students’ metaphorical reasoning. Part of the success of the instructional method relates to its ability to help students reason about, assess, and communicate about the logical structure of mathematical activity. In the case presented, this entailed a students’ shift from using properties to describe examples to using examples to relate various properties. The metaphor thus imbued key example sequences with meta-theoretical significance. We introduce the term “wedge” to describe such examples that distinguish oft-conflated properties. We also present our analytical criteria for empirically verifying the specific influence of the metaphorical aspect of instruction.

A Dialogic Method of Presenting Proofs: Focus on Fermat’s Little Theorem

Boris Koichu and Rina Zazkis

Abstract: Twelve participants were asked to decode a proof of Fermat’s Little Theorem and present it in a form of a script for a dialogue between two characters of their choice. Our analysis of these scripts focuses on issues that the participants identified as ‘problematic’ in the proof and on how these issues were addressed. Affordances and limitations of this dialogic method of presenting proofs are exposed, by means of analyzing how the students’ correct, partial or incorrect understanding of the elements of the proof are reflected in the dialogues. The difficulties identified by the participants are discussed in relation to past research on undergraduate students’ difficulties in proving and in understanding number theory concepts.

On the Role of Pedagogical Content Knowledge in Teachers’ Understanding of Commutativity and Associativity

Steven Boyce

Abstract: The purpose of this study is to investigate a relationship between mathematical content knowledge and pedagogical knowledge of content and students (Hill, Ball, & Shilling, 2008), in the context of algebra. As participants in a paired teaching experiment, mathematics education doctoral students revealed their understandings of commutativity and associativity (cf. Larsen, 2010). Although the participants’ knowledge of children’s initial understandings of algebra and familiarity with mathematics education literature influenced their own mathematics reasoning, the difficulties they encountered were similar to those of undergraduates without such pedagogical content knowledge.

Developing Facility with Sets of Outcomes by Solving Smaller, Simpler Counting Problems

Elise Lockwood

Abstract: Combinatorial enumeration has a variety of important applications, but there is much evidence indicating that students struggle with solving counting problems. In this paper, the use of the problem-solving heuristic of solving smaller, similar problems is tied to students’ facility with sets of outcomes. Drawing upon student data from clinical interviews in which post-secondary students solved counting problems, evidence is given for how numerical reduction of parameters can allow for a more concrete grasp of outcomes. The case is made that the strategy is particularly useful within the area of combinatorics, and avenues for further research are discussed.

Pre-Service Secondary Teachers’ Meanings for Fractions and Division

Cameron Byerley and Neil Hatfield

Abstract: In this study, seventeen math education majors completed a test on fractions and quotient. From this group, one above-average calculus student was selected to participate in a six-lesson teaching experiment. The major question investigated was “what constrains and affords the development of the productive meanings for division and fractions articulated by Thompson and Saldanha (2003)?” The student’s thinking was described using Steffe and Olive’s (2010) models of fractional knowledge. The report focuses on the student’s part-whole meaning for fractions and her difficulty assimilating instruction on partitive meanings for quotient. Her part-whole meaning for fractions led to the resilient belief that any partition of a length of size m must result in m, unit size pieces. It was non-trivial to develop the basic meanings underlying the concept of rate of change, even with a future math teacher who passed calculus.

Preservice Teachers’ Mathematical Knowledge for Teaching and Concepts of Teaching Effectiveness:
Are They Related?

Jathan Austin

Abstract: Mathematical knowledge for teaching (MKT) is essential for effective teaching of elementary mathematics. Given the importance of MKT, MKT and conceptions of teaching effectiveness should not develop independently. The purpose of this study was to examine whether and how K-8 pre-service teachers’ MKT and personal mathematics teacher efficacy beliefs are related. Results indicated overconfidence in teaching ability was prevalent, with the majority of participants exhibiting a strong sense of personal mathematics teacher efficacy but low levels of MKT. Pre-service teachers with high levels of MKT, however, reported a more accurate assessment of their teaching effectiveness. Results also indicated that examining pre-service teachers’ self-evaluations of MKT is helpful for understanding pre-service teachers’ personal mathematics teacher efficacy beliefs. Moreover, the results of this study point to the inadequacies of existing measures of teacher efficacy beliefs that do not parse out differences in efficacy beliefs according to a number of contextual factors.

Students’ Emerging Understandings of the Polar Coordinate System

Teo Paoletti, Kevin Moore, Jackie Gammaro, and Stacy Musgrave

Abstract: The Polar Coordinate System (PCS) arises in a multitude of contexts in undergraduate mathematics. Yet, there is a limited body of research investigating students’ understandings of the PCS. In this report, we discuss findings from a teaching experiment concerned with exploring four pre-service teachers’ developing understandings of the PCS. We illustrate the role students’ meanings for angle measure played while constructing the PCS. Specifically, students with a stronger understanding of radian angle measure more fluently constructed the PCS than their counterparts. Also, we found that various aspects of the students’ understandings of the Cartesian coordinate system (CCS) became problematic as they transitioned to the PCS. For instance, mathematical differences between the polar pole and Cartesian origin presented the students difficulties. Collectively, our findings highlight important understandings that can support or prevent students from developing a robust conception of the PCS.

Understanding Mathematical Conjecturing

Jason Belnap and Amy Parrott

Abstract: In this study, we open up discussions regarding one of the unexplored aspects of mathematical sophistication, the inductive work of conjecturing. We consider the following questions: What does conjecturing entail? How do the conjectures of experts and novices differ? What characteristics, behaviors, practices, and viewpoints distinguish novice from expert conjecturers? and What activities enable individuals to make conjectures? To answer these questions, we conducted a qualitative research study of eight participants at various levels of mathematical maturity. Answers to our research questions will begin to provide an understanding about what helps students develop the ability to make mathematical conjectures and what characteristics of tasks and topics may effectively elicit such behaviors, informing curriculum development, assessment, and instruction.

Building Knowledge for Teaching Rates of Change:
Three Cases of Physics Graduate Students

Natasha Speer and Brian Frank

Abstract: Over the past two decades education researchers have demonstrated that various types of knowledge, including pedagogical content knowledge, influence teachers' instructional practices and their students’ learning opportunities. Findings suggest that by engaging in the work of teaching, teachers acquire knowledge of how students think, but we have not yet captured this learning as it occurs. We examined whether novice instructors can develop such knowledge via the activities of attending to student work and we identified mechanisms by which such knowledge development occurs. Data come from interviews with physics graduate teaching assistants as they examined and discussed students' written work on problems involving rates of change. During those discussions, some instructors appear to develop new knowledge–either about students’ thinking or about the content—and others did not. We compare and contrast three cases representing a range of outcomes and identify factors that enabled some instructors to build new knowledge.

Undergraduate Students’ Models of Curve Fitting

Shweta Gupta

Abstract: The Models and Modeling Perspectives (MMP) has evolved out of research that began 26 years ago. MMP research uses Model Eliciting Activities (MEAs) to elicit students’ models of mathematical concepts. In this study MMP were used as conceptual framework to investigate the nature of undergraduate students’ models of curve fitting. Participants of this study were prospective mathematics teachers enrolled in an undergraduate mathematics problem solving course. Videotapes of the MEA session, class observation notes, and anecdotes from class discussions served as the sources of data for this study. Iterative videotape analyses as described in Lesh and Lehrer (2003) were used to analyze the videotapes of the participants working on the MEA. Results of this study discuss the nature of students’ models of the concept of curve fitting and add to the introductory undergraduate statistics education research by investigating the learning of the topic curve fitting.

Perspectives that Some Mathematicians Bring to University Course Materials Intended for Prospective Elementary Teachers

Elham Kazemi and Yvonne Lai

Abstract: Many elementary teachers receive their certification in undergraduate contexts, where they are taught mathematics content courses by mathematics faculty. However, there is a disconnect between courses typically taught in mathematics department, such as service courses for engineering or advanced mathematics courses, and mathematics content courses for teachers – often both in the instructors' experience with the material as well as in the way the courses are taught. In this paper, we report on four mathematicians' reviews of one set of materials for content courses for prospective elementary teachers. We report on perspectives these mathematicians brought to the materials regarding mathematics, mathematical knowledge for teaching, and teaching. We report on analysis of these perspectives for what may be visible and invisible about mathematical knowledge for teaching and the work of teaching.

On the Sensitivity of Problem Phrasing – Exploring the Reliance of Student Responses on Particular Representations of Infinite Series

Danielle Champney

Abstract: This study will demonstrate the ways in which students’ ideas about convergence of infinite series are deeply connected to the particular representation of the mathematical content, in ways that are often conflicting and self-contradictory. Specifically, this study explores the different limiting processes that students attend to when presented with five different phrasings of a particular mathematical task - ∑(1/2)^n - and the ways in which each phrasing of the task brings to light different ideas that were not evident or salient in the other phrasings of the same task. This research suggests that when attempting to gain a more robust understanding of the ways that students extend the ideas of calculus – in this case, limit – one must take care to attend to not only students’ reasoning and explanation, but also the implications of the representations chosen to probe students’ conceptions, as these representations may mask or alter student responses.