Conference Program/Abstracts

Abstracts - Preliminary Reports

Wait a Minute…Is That Enough to Make a Difference?

Daniel Reinholz and Mary Pilgrim

Abstract: The “one-minute paper” (Stead, 2005) is a technique for facilitating communication between students and the teacher and promoting reflection. In this paper we focus on the types of questions students ask and how they may be related to success. We present preliminary results from an introductory university-level calculus course, indicating that the nature of questions asked by more successful and less successful students are different, suggesting that the types of reflections that students engage in may have a significant impact on the efficacy of such an intervention.

Rethinking Business Calculus in the Era of Spreadsheets

Mike May

Abstract: The author is Writing and electronic “book” to support the teaching of calculus to business students with the assumption that they will use a spreadsheet as their main computational engine. With the change in technology, it is appropriate to rethink the content of the course as a different technology makes different tasks accessible. This study looks at what the content of the course should be. It compares the official learning objects of the course, with de facto learning objectives obtained by analyzing final exams from 20 sections of the course, and with the results of a survey of the faculty of the business school, the client discipline. It is intended that this preliminary study establishes a baseline that can be used to evaluate the effectiveness of the new approach to the course.

Expert Performance on Routine and Novel Integral Application (Volume) Problems

Krista Toth and Vicki Sealey

Abstract: Past research has shown that students struggle when applying the definite integral concept, and these difficulties stem from incomplete understanding of the integral’s underlying structure. This study aims to provide insight into the construction of effective mental structures for integrals by examining experts’ solutions to volume problems. Seven mathematics faculty members from a large, public university solved three calculus-level volume problems (two routine, one novel) in videotaped interview sessions. Preliminary analysis shows that the experts have a rich understanding of definite integrals, and the few instances of errors seemed to be a result of inattention as opposed to a deficit in understanding. Their problem-solving process was highly structured and detailed. The experts’ visual representations varied from sparse and static to fully 3-dimensional and dynamic. We hope to use this and past student data to construct a framework for analyzing student understanding of integral volume problems.

How Pre-Service Teachers in Content Courses Revise Their Mathematical Communication

Nina White

Abstract: Math content courses aim to develop mathematical reasoning and communication skills in future teachers. Instructors often assign problems requiring in-depth written explanations to develop these skills. However, when a student’s conception is incorrect, does written feedback from the instructor create the cognitive dissonance necessary to effect realignment of the student’s understanding? These conceptions may be mathematical (“what is a fraction?”) or meta-mathematical (“what constitutes a justification?”). Assigning problem revisions theoretically creates space for cognitive dissonance by having students rethink their solutions. I investigate a revision assignment in a course for future teachers to understand the nature of students’ revisions and the possible impetuses for these revisions. In particular, I find preliminary evidence that students’ revisions demonstrate changes in their language, mathematics, and use of examples and representations. Further, students’ adoption of new representations in their solutions are largely due to observing peers’ presentations rather than to instructor feedback.

Opportunity to Learn from Mathematics Lectures

Emilie Wiesner, Tim Fukawa-Connelly, and Aaron Weinberg

Abstract: Many mathematics students experience proof-based classes primarily through lectures, although there is little research describing what students actually learn from such classroom experiences. Here we outline a framework, drawing on the idea of the implied observer, to describe lecture content; and apply the framework to a portion of a lecture in an abstract algebra class. Student notes and interviews are used to investigate the implications of this description on students' opportunities to learn from proof-based lectures. Our preliminary findings detail the behaviors, codes, and competencies that an algebra lecture requires. We then compare those with how students behave in response to the same lecture with respect to sense-making and note-taking, and thereby how they approach opportunities to learn.

Assessment of Students’ Understanding of Related Rates Problems

Costanza Piccolo and Warren Code

Abstract: This study started with a thorough analysis of student work on problems involving related rates of change in a first-year differential calculus course at a large, research-focused university. In two sections of the course, students' written solutions to geometric related rates problems were coded and analyzed, and students' learning was tracked throughout the term. Three months after the end of term, "think-aloud'' interviews were conducted with some of the students who completed the course. The interviews and some of the written assessments were structured based on the classification of key steps in solving related rates proposed by Martin (2000). Our preliminary findings revealed a widespread, persistent use of algorithmic procedures to generate a solution, observed in both the treatment of the physical and geometric problem, and the approach to the differentiation, and raised the question of whether traditional exam questions are a true measure of students' understanding of related rates.

Not All Informal Representations are Created Equal

Kristen Lew, Juan Pablo Mejia-Ramos, and Keith Weber

Abstract: Some mathematics educators and mathematicians have suggested that students should base their proofs on informal reasoning (Garuti et al. 1998). However, the ways in which students implement informal representations are not well understood. In this study, we investigate informal representations made by undergraduates during proof construction. Their use of informal representations will be compared to mathematicians’ use of informal representations as described in Alcock (2004) and Samkoff et al. (2012). Further, an analysis of different types of informal representations will investigate the necessity to treat these different representations more carefully in the future.

Fostering Students’ Understanding of the Connection Between Function and Derivative: A Dynamic Geometry Approach

Dov Zazkis

Abstract: Students’ difficulties with relating the graphs of functions to the graphs of their derivatives have been well documented in the literature. Here I present a Geometer’s Sketchpad based applet, which was used as part of a technologically enriched Calculus I course. Individual interviews with students conducted after this in-class activity show evidence of varied and powerful student problem solving strategies that emerged after participation in the activity.

Determining the Structure of Student Study Groups

Gillian Galle

Abstract: Although students are expected to spend time outside the classroom furthering their understanding of the material, there has been little verification of what students actually do when they study. This project observed undergraduate students studying together outside of the classroom setting in order to determine what study groups formed and what structures described these groups. Elements of social network analysis were employed to identify the groups that students formed. Transcripts of the study sessions were coded and frequency counts were established for each type of student interaction in order to characterize the roles students assumed while studying. This paper discusses the process of identifying the study groups and sets the groundwork for sharing the student roles. One main finding of this work is that the presence of a student recognized as an authority or facilitator of the group impacts the type of

conversations that occur in the group setting.

Assessing Pre-Service Teachers’ Conceptual Understanding of Mathematics Using Praxis II Data

Revathi Narasimhan

Abstract: We summarize the preliminary results of a study of conceptual understanding of mathematics by pre-service secondary school math teachers. Our research involves the statistical analysis of data from an actual mathematics Praxis II licensure exam, which was administered nationwide. Through a quantitative, item by item analysis, using a classification of these test items by conceptual difficulty, we obtain insight into the conceptual issues that pre-service teachers have great difficulty with. Our preliminary results show a significant gap between computational and abstract mathematical processes. This in turn, affects the ability of pre-service teachers to be fluent in the domains of both subject and pedagogical content knowledge.

A Case Study on a Diverse College Algebra Classroom:
Analyzing Pedagogical Strategies to Enhance Students’ Mathematics Self-Efficacy

Michael Furuto and Derron Coles

Abstract: Shifting demographics show America rapidly diversifying, yet research indicates that an alarming number of diverse students continue to struggle to meet learning outcomes of collegiate mathematics curriculum. Consequently, recruitment and retention of diverse students in STEM majors is a pervasive issue. Using a sociocultural perspective, this study examined the effect of two pedagogical strategies (traditional instruction and cooperative learning) in a diverse College Algebra course on enhancing students’ mathematics self-efficacy. Particular attention was paid to investigating the role student discourse and interaction play in facilitating learning, improving conceptual understanding, and empowering students to engage in future self-initiated communal learning. The goal is to develop an effective classroom model that cultivates advancement in content knowledge and enculturation into the STEM community, culminating in a higher retention rate of diverse students in STEM. Preliminary data analysis suggests that a hybrid model encompassing both traditional instruction and cooperative learning successfully enhances students’ self-efficacy.

A Multidimensional Analysis of Instructional Practices

Melissa Mills

Abstract: This study is an investigation of the questions that are asked by four faculty members who were teaching advanced mathematics. Each question was analyzed along three dimensions: the expected response type of the question, the Bloom’s Taxonomy level, and the context of the question within the mathematics content.

Crossing Community Boundaries: Collaboration Between

Mathematicians and Mathematics Educators

Sarah Bleiler

Abstract: Effective mathematics teachers are able to make connections between mathematical content and pedagogy in their professional practice. One of the most readily prescribed approaches for facilitating teachers’ ability to make such connections is through the development of collaborations between mathematicians and mathematics educators in venues related to teacher professional development. Most prior research related to collaborative endeavors between these two groups has focused on the products, rather than the process, of collaboration. In this preliminary research report, I present the results of an interpretative phenomenological case study that investigated the team-teaching experiences of a mathematician and a mathematics educator within the context of an undergraduate mathematics teacher preparation program. I present extracts from interviews that highlight the instructors’ perceptions related to crossing the boundaries of their professional communities of practice, and engage participants in discussion about relevant “boundary crossing” in their own institutional contexts.

Interplay Between Concept Image and Concept Definition: Definition of Continuity

Gaya Jayakody

Abstract: This study looks at the interplay between the concept image and concept definition when students are given a task that requires direct application of the definition of continuity of a function at a point. Data was collected from 37 first year university students. It was found that different students apply the definition to different levels, which varied from formal deductions (based on the application of the definition) to intuitive responses (based on rather loose and incomplete notions in their concept image).

An Examination of Proving Using a Problem Solving Framework

Milos Savic

Abstract: A link between proving and problem solving has been well established in the literature (Furinghetti & Morselli, 2009; Weber, 2005). In this paper, I discuss similarities and differences between proving and problem solving by using the Multidimensional Problem-Solving Framework created by Carlson and Bloom (2005) on Livescribe pen data from a study of proving (Author, 2012). I focus on two participants’ proving processes: Dr. G, a topologist, and L, a mathematics graduate student. Many similarities were revealed by using the Carlson and Bloom framework, but also some differences distinguish the proving process from the problem-solving process. In addition, there were noticeable differences between the proving of the mathematician and the graduate student. This study may influence a proving-process framework that can encompass both the problem-solving aspect of proving and the differences found.

Talking Mathematics: An Abstract Algebra Professor’s Teaching Diaries

Sepideh Stewart, John Paul Cook, Ralf Schmidt, and Ameya Pitale

Abstract: The world of a mathematician, with all its creativity and precision is fascinating to most people. This study is an account of collaboration between mathematicians and mathematics educators. In order to examine a mathematician’s daily activities, we have primarily employed Schoenfeld’s goal-orientated decision making theory to identify his Resources, Orientations and Goals (ROGs) in teaching an abstract algebra class. Our preliminary results report on a healthy and positive atmosphere where all involved freely express their views on mathematics and pedagogy.

Scaling Up Reinvention: Developing a Framework for Instructor Roles in the Classroom

Jungeun Park, Jason Martin, and Michael Oehrtman

Abstract: Studies have shown that students have difficulty with the concept of limit, especially when reasoning about formal limit definitions. We conducted a five-day teaching experiment (TE) in a second semester calculus classroom in which students were asked to reinvent a formal sequence convergence definition. Author 3 (2011) detailed how pairs of students reinvented sequence convergence definitions but did not attempt the same instructional heuristic in the classroom. Our analysis focused on the instructor prompts and the TE students' subsequent group discussion through their use of key words and visuals in revising their definition. An interview with the instructor was conducted to investigate his intention of using specific prompts and his thinking about the TE group's choice of words and visuals. In our preliminary analysis, we found that the roles of the instructor were extended beyond those roles previously reported as roles for facilitators with pairs of students.

An Analysis of First Semester Calculus Students’ Use of Verbal and Written Language When Describing the Intermediate Value Theorem

Vicki Sealey and Jessica Deshler

Abstract: This preliminary report describes the second stage of data collection and analysis in a larger study that examines students’ written and verbal language when studying basic theorems in a first-semester calculus course. We examine students’ difficulties with understanding and using mathematical language and notation in both formal written work and informal verbal descriptions. Not surprisingly, the students in our study rarely use formal mathematical language without being prompted to do so. One surprising result was that while many students do understand the mathematical notation in the theorems, and can illustrate this graphically when prompted, they still do not use this notation when providing their own written (or verbal) description of a theorem. Preliminary results suggest that our biggest obstacle as teachers is not in getting our students to understand the notation, but instead lies in convincing our students of the power that comes from this notation in describing a concept, thus encouraging our students to use this notation in their own written work.

Comparing a “Flipped” Instructional Model in an undergraduate Calculus III Course

Nicholas Wasserman, Scott Norris and Thomas Carr

Abstract: In this small comparative study, we explore the impact of “flipping” the instructional delivery of content in an undergraduate Calculus III course. Two instructors collaborated to determine daily content and lecture notes; one instructor altered the instructional delivery of the content (not the content itself), utilizing videos to communicate procedural course content to students out-of-class, with time in-class spent on conceptual activities and homework problems. With similar numbers (n=41 and n=40) and types of students in each class, student performance on tests for both classes will be compared to determine any significant differences in achievement related to “flipping” the instructional delivery of content.

Development and Analysis of a Basic Proof Skills Test

Sandra Merchant and Andrew Rechnitzer

Abstract: We have developed a short (16 question) basic skills test for use in our institution's transition-to-proof course that assesses basic skills required to succeed in such a course. Using this test in our core introductory proof course, we have found that students are generally deficient in a number of skills assumed by instructors. In addition, using this test as a pre/post-test we have found that in this course students are learning some concepts well, but that learning gains on other concepts are much below desired levels. Finally, administration of the test to students in a higher level course has allowed us to assess retention of these skills. At this preliminary stage these skills appear to be retained into higher-level proof courses, but more data collection is needed, as well as a more extensive instrument to assess proof skills, rather than simply basic logic and comprehension.

A Microgenetic Study of One Students’ Sense Making About the Temporal Order of Delta and Epsilon

Aditya Adiredja

Abstract: The formal definition of a limit, or the epsilon delta definition is a critical topic in calculus for mathematics majors’ development and the first chance for students to engage with formal mathematics. This report is a microgenetic study of one student understanding of the formal definition focusing on a particularly important relationship between epsilon and delta. diSessa’s Knowledge in Pieces and Knowledge Analysis provide frameworks to explore in detail the structure of students’ prior knowledge and their role in learning the topic. The study documents the progression of the student’s claims about the dependence between delta and epsilon and explores relevant knowledge resources.

An Investigation of Pre-Service Secondary Mathematics Teachers’ Development and Participation in Argumentation

Lisa Rice

Abstract: This study investigates how two professors and pre-service secondary mathematics teachers engage in argumentation and proof in two courses. One course under investigation is a geometry course; the second is a methods of teaching mathematics course. The research also studies the how professors and pre-service teachers construct arguments and proofs. Examining the classroom discourse to understand how it may impact argumentation practices is another aspect of the research. Case study and grounded theory approaches are used to guide the data collection and analysis. Some data collected include interviews with the two professors and pre-service teachers and observations of the two courses and the pre-service teachers’ classrooms during their student teaching. Data analysis so far indicates the geometry professor engages students in argumentation and proof in multiple ways.

Emergent Modeling and Riemann Sum

Kritika Chhetri and Jason Martin

Abstract: This research focuses on mental challenges that students face and how they resolve these challenges while transiting from intuitive reasoning to constructing a more formal mathematical structure of Riemann sum while modeling “real life” contexts. A pair of Calculus I students who had just received instruction on definite integral defined using Riemann sums and illustrated as area under the curve participated in multiple interview sessions. They were given contextual problems related to Riemann sums but were not informed of this relationship. Our intent was to observe students’ transitioning from model of to model for reasoning while modeling these problem situations. Results indicate that students conceived of five major conceptions during their first task and their reasoning from the first task that became a model for reasoning about their next task. In this paper we detail those conceptions and their reasoning that became model for reasoning on the second task.

Students’ Way of Thinking About Derivative and its Correlation to Their Ways of Solving Applied Problems

Shahram Firouzian

Abstract: Previous researchers have examined students’ understanding of derivative and their difficulties in solving applied problems and/or their difficulties in applying the basic knowledge of derivative in different contexts. There has not been much research approaching students’ ways of thinking about derivative through the lens of applied questions. In this research, first I categorized the students’ way of thinking about the basic concept of derivative by running a survey of questions addressing the different ways of thinking about derivative based on the existing research works. While analyzing these surveys, I used grounded theory and added more ways of thinking about derivative. I specially noticed very incomplete ways of thinking about derivative as described below. Since my goal was looking at the students’ ways of thinking about derivative through the lenses of applied questions, I also piloted my applied questions survey with 51 multivariable calculus students. I noticed a lot of students struggling with defining variables (the initial translation as described below) and if they could define the variable, a lot of them struggled on applying their ways of thinking about derivative into solving the applied problem. These difficulties are great venues to study their ways of thinking about derivative using their struggle in the applied questions. This is a summary of my initial works on this ongoing research, the goal of which is to shed new insights into students’ solving of applied problems.

Computational Thinking in Linear Algebra

Spencer Bagley and Jeff Rabin

Abstract: In this work, we examine students' ways of thinking when presented with a novel linear algebra problem. We have hypothesized that in order to succeed in linear algebra, students must employ and coordinate three modes of thinking, which we call computational, abstract, and geometric. This study examines the solution strategies that undergraduate honors linear algebra students employ to solve the problem, the variety of productive and reflective ways in which the computational mode of thinking is used, and the ways in which they coordinate the computational mode of thinking with other modes.

Pre-service Secondary Mathematics Teachers’ Statistical Preparation:
Interpreting the News

Joshua Chesler

Abstract: Undergraduate mathematics programs must prepare teachers for the challenges of teaching statistical thinking as advocated in standards documents and statistics education literature. This preliminary report presents initial results from a study of pre-service secondary mathematics teachers at the end of their undergraduate educations. Although nearly all had completed a required upper-division statistics course, most were challenged by two tasks which required a critical analysis of the use of statistics in newspaper articles. Some patterns emerged in the incorrect answers, including a tendency to focus on potential sampling issues which were not relevant to the tasks. The session will explore the nature and sources of these difficulties with statistical thinking and statistical communication and it will explore the implications for undergraduate mathematics and statistics teacher preparation.

The Effects of Formative Assessment on Students’ Zone of Proximal Development in Introductory Calculus

Rebecca Dibbs and Michael Oehrtman

Abstract: One of the challenges of teaching introductory calculus is the large variance in student backgrounds. Formative assessment can be used to target which students need help, but little is known about why formative assessment is effective with adult learners. The purpose of this qualitative study was to investigate which functions of formative assessment help instructors to provide the scaffolding needed to help students in an introductory calculus course progress through their Zones of Proximal Development during the weekly group labs. By providing students a low-stakes opportunity to demonstrate their current understanding, students were able to evaluate their progress and ask further questions after the activity was completed; this information was used to plan the discussion in the next class period. This discussion provided the scaffolding students needed to progress through the activities as well as providing peripheral participation opportunities for students who would not ordinarily ask questions during class.

Verifying Trigonometric Identities: Proof and Students’ Perceptions of Equality

Benjamin Wescoatt

Abstract: This preliminary study explores how students’ perceptions of the equality of trigonometric expressions evolve during the process of verifying trigonometric identities (VTI). If students already view the purported equality as being true, VTI may not offer much in the way of learning experiences for students. Using a semiotic perspective to analyze student work, this study attempts to describe the evolution of students’ perceptions of identities upon application of VTI, focusing on the components of the student’s VTI process that contribute to the evolution. Initial analyses of interviews conducted while students proved identities indicate that students are not fully convinced that the identities are initially true. However, successful VTI, signaled, for example, by the use of an idiosyncratic equality construction, endows equality on not only the purported identity but on ancillary equality statements generated as part of the VTI process.

Student Responses to Team Based Learning in Tertiary Mathematics Courses

Judy Paterson, Louise Sheryn, and Jamie Sneddon

Abstract: Starting in 2009 we have implemented a Team Based Learning (TBL) model of delivery in two mathematics courses and one mathematics education course involving a total of 295 students. Qualitative data from evaluations, observations and interviews is used to begin to answer four questions raised by the Seldens (2001) regarding teaching mathematics at tertiary levels. Our analysis indicates that students say that TBL creates an environment in which they are active, have productive arguments and discussions and benefit from immediate feedback. There is scant evidence of any group being disadvantaged by this model of delivery.

Paradoxes of Infinity – The Case of Ken

Chanakya Wijeratne

Abstract: Previous studies have shown that the normative solutions of the Pin-Pong Ball Conundrum and the Pin-Pong Ball Variation are difficult to understand even for learners with advanced mathematical background such as doctoral students in mathematics. This study examines whether this difficulty is due to the way they are set in everyday life experiences. Some variations of the Pin-Pong Ball Conundrum and the Pin-Pong Ball Variation and their abstract versions set in the set theoretic language without any reference to everyday life experiences were given to a doctoral student in mathematics. Data collected suggest that the abstract versions can help learners see beyond the metaphorical language of the paradoxes. The main contribution of this study is revealing the possible negative effect of the metaphorical language of the paradoxes of infinity on the understanding of the learner.

Self-inquiry in the Context of Undergraduate Problem Solving

Todd Grundmeier, Dylan Retsek and Dara Stepanek

Abstract: Self-inquiry is the process of posing questions to oneself while solving a problem. The self-inquiry of thirteen undergraduate mathematics students was explored via structured interviews requiring the solution of both mathematical and non-mathematical problems. The students were asked to verbalize any thought or question that arose while they attempted to solve a mathematical problem and its nonmathematical logical equivalent. The thirteen students were volunteers who had each taken at least four upper division proof-based mathematics courses. Using transcripts of the interviews, a coding scheme for questions posed was developed and all questions were coded. While data analysis of the posed questions is ongoing, initial analysis suggests that the “good” mathematics students focus more questions on legitimizing their work and fewer questions on specification of the problem-solving task. Additionally, the self-inquiry of “fast” problem solvers mirrored that of the strong students with even less focus on specification questions.

Systematic Intuitive Errors on a Prove-or-Disprove Monotonicity Task

Kelly Bubp

Abstract: Despite the importance of intuitive and analytical reasoning in proof tasks, students have various difficulties with both types of reasoning. Such difficulties may be attributed to insufficient intuition, logical reasoning skills, or concept images. However, dual-process theory asserts that intuition can form faulty representations of tasks based on systematic errors before analytical reasoning can respond. Thus, students' difficulties could be attributed to systematic intuitive errors rather than inadequate intuitive or analytical reasoning. In this study, I conducted task-based interviews with four undergraduate and one graduate mathematics major in which they completed prove-or-disprove tasks. In this paper, I discuss the systematic intuitive errors committed by these students on a monotonicity task. These errors led all five students to believe incorrectly that the statement in the task was true. Furthermore, each student engaged in correct mathematical reasoning guided by their incorrect intuitive representations.

A Coding Scheme for Analyzing Graphical Reasoning on Second Semester Calculus Tasks

Rebecca Schmitz

Abstract: As a first step in studying students’ spatial reasoning ability, preference, and their impact on performance in second semester calculus, I ran a pilot study to develop interview tasks and a coding scheme for analyzing the interviews. Four videotaped interviews were conducted with each of the five participants and the video was coded for graphical reasoning. I will discuss my coding scheme and share some preliminary results. I hypothesize that the coding scheme may help identify a student’s preference and ability for spatial reasoning.

Mathematician’s Tool Use in Proof Construction

Melissa Goss, Jeffrey King, and Michael Oehrtman

Abstract: The goals on teaching proof are to “help students develop an understanding of proof that is consistent with that shared and practiced by mathematicians of today.” This study sought to describe the tools and reasoning techniques used by mathematicians to construct and write proofs. Task-based clinical interviews were conducted with 3 research mathematicians in varying research fields. The tasks were upper-undergraduate and lower-graduate level proofs from linear algebra, basic analysis, and abstract algebra. Data were coded based on a framework constructed from Dewey’s theory of inquiry and the characterizations of conceptual insight and technical handle. Preliminary results indicate the task of discovering a conceptual insight that can potentially lead to a proof can be problematic, and there are distinct moments in the construction process when the problem changes from “why should this be true?” to “how can I prove that?”

Initial Undergraduate Student Understanding of Statistical Symbols

Samuel Cook and Tim Fukawa-Connelly

Abstract: In this study we use the tradition of semiotics to motivate an exploration of the knowledge of, and facility with, the symbol system of statistics that students bring to university. We collected a sample of incoming mathematics majors in their first semester of study, prior to taking any statistics coursework and engaged each in a task-based interview using a think-aloud protocol with questions designed to assess their fluency with basic concepts and symbols of statistics. Our findings include that students find symbols arbitrary and difficult to associate with the concepts. Second, that generally, no matter the amount of statistics that students took in high school, including Advanced Placement courses, they generally have relatively little recall of topics. Most can calculate the mean, median and mode, but they generally remember little beyond that. Finally, students have difficulty connecting practices or procedures to meaning.

Proof Structure in the Context of Inquiry Based Learning

Alyssa Eubank, Shawn Garrity and Todd Grundmeier

Abstract: Data was collected from the final exams of 68 students in three sections of an introductory proofs course taught from an inquiry-based perspective. Inquiry-based learning (IBL) gives authority to students and allows them to present to their peers, rather than the instructor being the focus of the class and the authority on proof. This data was analyzed with a focus on proof structure. The selected final exam problems included concepts that were introduced prior to the course and others that were new to students. This research utilizes an adaptation of Toulmin’s method for argumentation analysis. Our goal was to compare the proof structures generated by these students to previous research that also applied some form of Toulmin’s scheme to mathematical proof. There was significant variety of proof structures, which could be a result of the IBL atmosphere.

Analyzing Calculus Concept Inventory Gains in Introductory Calculus

Matthew Thomas and Guadalupe Lozano

Abstract: Research in science education, particularly physics education, indicates that students in Interactively-Engaged classrooms are more successful on tests of basic conceptual knowledge. Despite this, undergraduate mathematics courses are dominated by lectures in which students take a passive role. Given the value of such tests in assessing students' conceptual knowledge, the method for measuring such change is largely unexplored. In our study, students were given one such inventory, the Calculus Concept Inventory, in introductory Calculus classes as a pretest and posttest. We address issues of how gains might be measured on this instrument using two techniques, and the implications of using each of these measures.

A Modern Look at the Cell Problem

Jennifer Czocher

Abstract: Mathematical modeling perspectives continue to become viable lenses for examining students' mathematical thinking in novel contexts. Thus, it is vital to re-visit past foundational work in problem solving in order to connect new ideas and interpretations with accepted knowledge. The objective of this paper is to examine results from a well-known problem setting (The Cell Problem) to explore alternative interpretations of students' mathematical work.

Bringing the Familiar to the Unfamiliar: The Use of Knowledge from Different Domains in the Proving Process

Kathleen Melhuish

Abstract: This report considers student proof construction in small groups within an inquiry-orientated abstract algebra classroom. During an initial analysis, several cases emerged where students used familiar knowledge from another mathematical domain to provide informal intuition. I will report on two episodes in order to illustrate how this intuition could potentially aid or hinder the construction of a valid proof.

Investigating Student Understanding of Eigentheory in Quantum Mechanics

Warren Christensen

Abstract: An initial investigation into students’ understanding of Eigen theory using semi-structured interviews was conducted with students at the end of a first-semester course in quantum mechanics. Many physics faculty would expect students to have mastery of basic matrix multiplication after a course in Linear Algebra, and especially so after fairly extensive use of matrices in quantum mechanics in the context of Ising model spin problems. Using a previously published interview protocol by Henderson et al, student reasoning patterns were investigated to probe to what extent there reasoning patterns were similar to those identified among Linear Algebra students. Reasoning patterns appeared quite consistent with previous work; that is, students used superficial algebraic cancellation, and demonstrated difficulty interpreting their result even when they arrived at a correct solution. The interview protocol was modified slightly to probe whether or not students felt the tasks they were engaging in were mathematical or physics-related. Additional questions were added at the end of the protocol about how these concepts were used in their quantum mechanics course. Students were somewhat successful relating them to Hamiltonians and energy eigenvalues, but couldn’t articulate the type of physical situations where they might be useful.

Student Difficulties Setting Up Statistics Simulations in Tinkerplots�

Erin Glover and Jennifer Noll

Abstract: This preliminary report addresses the need for research that explores how technology changes the way students think about statistics and the ways technology can be used to enable students to construct models to solve statistical problems. This study focuses on student challenges interpreting a single trial of a statistical experiment and setting up TinkerplotsTM simulations. Sixteen students in a lower division introductory statistics course worked on a task involving the "One Son Policy", a situation in which families continue to have children until they have a boy. Students’ interpretations of what a single trial represented in the One Son activity fell into four categories, three of which were completing the task. Further, students had difficulty with using the technology to set up and interpret a simulation to address the question. These results suggest that the process of setting up a computer simulation to answer a statistical question is quite complex.

Providing Opportunities for College-Level Calculus Students to Engage in Theoretical Thinking

Dalia Challita and Nadia Hardy

Abstract: Previous research has reported an absence of a theoretical thinking component in college-level Calculus courses; moreover, valid arguments can be made for or against the necessity and feasibility of incorporating such a component. Our belief is, however, that students who wish to engage in theoretical thinking should be given the chance to do so in such a course. The current report presents a preliminary analysis of a study we conducted in a Calculus class in which we presented students with tasks, in the form of quizzes, intended to provoke a type of behavior that is indicative of theoretical thinking. Using Sierpinska et al.’s (2002) model as a basis for theoretical thinking we show that students were indeed engaged in theoretical thinking through these tasks. Our preliminary analysis of the results suggests that despite constraints often faced by instructors of such courses, incorporating such a component is indeed feasible.

Students’ Knowledge Resources About the Temporal Order of Delta and Epsilon

Aditya Adiredja and Kendrice James

Abstract: The formal definition of a limit, or the epsilon delta definition is a critical topic in calculus for mathematics majors’ development and the first chance for students to engage with formal mathematics. Research has documented that the formal definition is a roadblock for most students but has de-emphasized the productive role of their prior knowledge and sense making processes. This study investigates the range of knowledge resources included in calculus students’ prior knowledge about the relationship between delta and epsilon within the definition. diSessa’s Knowledge in Pieces provides a framework to explore in detail the structure of students’ prior knowledge and their role in learning the topic.

Two Students' Interpretation of Rate of Change in Space

Eric Weber

Abstract: This paper describes a model of the understandings of two first-semester calculus students, Brian and Neil, as they participated in a teaching experiment focused on exploring ways of thinking about rate of change of two-variable functions. I describe the students’ construction of directional derivative as they attempted to generalize their understanding of one-variable rate of change functions, and characterize the importance of quantitative and covariational reasoning in this generalization.

Conceptualizing Vectors in College Geometry:
A New Framework forAnalysis of Student Approaches and Difficulties

Oh Hoon Kwon

Abstract: This article documents a new way of conceptualizing vectors in college geometry. The complexity and subtlety of the construct of vectors highlight the need for a new framework that permits a layered view of the construct of vectors. The framework comprises three layers of progressive refinements: a layer that describes a global distinction between physical vectors and mathematical vectors, a layer that recounts the difference between the representational perspective and the cognitive perspective, and a layer that identifies ontological and epistemological obstacles in terms of transitions towards abstraction. Data was gathered from four empirical studies with ninety-eight total students to find evidence of the three major transition points in the new framework: physical to mathematical coming from the first layer, geometric to symbolic and analytic to synthetic from the second layer, and the prevalence of the analytic approach over the synthetic approach while developing abstraction enlightened by the third layer.

Jump Math Approach to Teaching Foundations Mathematics in 2-Year College Shows Consistent Gains in Randomized Field Trial

Taras Gula and Carolyn Hoessler

Abstract: Many first year college students struggle with foundational mathematics skills even after one semester of mathematics. JUMP math, a systematized program of teaching mathematics, claims that its approach, though initially designed for K-8, can strengthen skills at the foundations college math level as well. Students in sixteen sections of Foundations Mathematics at a college in Canada were randomly assigned to be taught with either the JUMP math approach or a typical teaching approach. Students were measure before and after on their competence (Wechsler test of Numerical Operations) and attitudes (Mathematics Attitudes Inventory) to identify any improvements. Results showed that students in JUMP classes had modest, but consistently higher improvements in competence when compared to students in non-JUMP classes, even after controlling for potential confounding variables, while improvements in Math Attitudes showed no differences.

Identifying Change in Secondary Mathematics Teachers’ Pedagogical Content Knowledge

Melissa Goss, Robert Powers, and Shandy Hauk

Abstract: Like several other research groups, we have been investigating multiple measures for capturing change in middle and high school teachers’ mathematical pedagogical content knowledge (PCK). This article reports on results among 14 teachers (of 16 enrolled) who have completed a virtual master’s program in mathematics education. The degree program seeks to develop content proficiency, cultural competence, and pedagogical expertise for teaching mathematics. Analysis included pre- and post-program data from classroom observations and written PCK assessment. Results indicate significant changes in curricular content knowledge on the observation instrument and significant changes in discourse knowledge on both the observation instrument and the written assessment. Additional path analysis suggests teacher discourse knowledge as measured by the written assessments is significantly related to discourse knowledge as measured by the post-program observation.

Difficulties in Using Variables – A Tertiary Transition Study

Ileana Borja-Tecuatl, Asuman Oktaç and María Trigueros

Abstract: This article describes the results obtained from a diagnostic instrument to establish the difficulties in understanding and using variables that engineering student’s have at the moment of their entrance to a public Mexican university, that does not examine the candidates prior to admittance. Once the difficulties were established, a 1-year treatment based on the 3 Uses of Variables Model was applied to foster a rich conception of variable in the students. The purpose of this treatment was on one hand to enrich the students’ concept of variable, to make it possible that they consider variables as dynamic objects that not only represent unknowns, but also describe relations between the objects they represent, and that may vary their usage along one same problem. On the other hand, we wanted to set the basis to study how a poor/rich conception of variable interferes with understanding the solution of a linear equations system.

Gestures: A Window to Mental Model Creation

Nancy Garcia and Nicole Engelke

Abstract: Gestures are profoundly integrated into communication. This study focuses on the impact that gestures have in a mathematical setting, specifically in an undergraduate calculus workshop. We identify two types of gesture – dynamic and static – and note a strong correlation between these movements and diagrams produced. Gesture is a primary means for students to communicate their ideas to each other, giving them a quick way to share thoughts of relative motion, relationships, size, shape, and other characteristics of the problem. Dynamic and static gestures are part of the students’ thinking, affecting how they view the problem, sway group thinking, and the construction of their diagrams.

Development of Students’ Ways of Thinking in Vector Calculus

Eric Weber and Allison Dorko

Abstract: In this talk, we describe the development of the ways of thinking of 25 vector calculus students over the course of one term. In particular, we characterize the generalizations that students made within and across interviews. We focus on the construction of the semi-structured pre and post interviews, trace the construction of explanatory constructs about student thinking that emerged from those interviews, and describe how those constructs fit within the broader literature on student thinking in advanced calculus. We conclude by exploring implications for future research and practical applications for educators.

Characteristics of Successful Programs in College Calculus: Pilot Case Study

Sean Larsen, Estrella Johnson and Steve Strand

Abstract: The CSPCC (Characteristics of Successful Programs in College Calculus) project is a large empirical study investigating mainstream Calculus 1 to identify the factors that contribute to success, to understand how these factors are leveraged within highly successful programs. Phase 1 of CSPCC entailed large-scale surveys of a stratified random sample of college Calculus 1 classes across the United States. Phase 2 involves explanatory case study research into programs that are successful in leveraging the factors identified in Phase 1. Here we report preliminary findings from a pilot case study that was conducted at a private liberal arts university. We briefly describe the battery of interviews conducted at the pilot site and discuss some of the themes that have emerged from our initial analyses of the interview data.

Cooperative Learning and Traversing the Continuum of Proof Expertise

Martha Byrne

Abstract: This paper describes preliminary results of a study aimed at examining the effects of working in cooperative groups on acquisition and development of proof skills. Particular attention will be paid to the varying tendencies of students to switch proof methods (direct, induction, contradiction, etc) based on their level of proof expertise.

Students’ Sense-Making in Mathematics Lectures

Aaron Weinberg, Tim Fukawa-Connelly, and Emilie Wiesner

Abstract: Many mathematics students experience proof-based classes primarily through lectures, although there is little research describing what students actually learn from such classroom experiences. Here we outline a framework, drawing on the idea of the implied observer, to describe lecture content; and apply the framework to a portion of a lecture in an abstract algebra class. Student notes and interviews are used to investigate the implications of this description on students' opportunities to learn from proof-based lectures. Our preliminary findings detail the behaviors, codes, and competencies that an algebra lecture requires. We then compare those with how students behave in response to the same lecture with respect to sense-making and note-taking, and thereby how they approach opportunities to learn.

Guided Reflections on Mathematical Tasks: Fostering MKT in College Geometry

Josh Bargiband, Sarah Bell and Tetyana Berezovski

Abstract: This study is a part of ongoing research on development of Mathematical Knowledge for Teaching (MKT) in mathematical content courses. Reflective practice represents a central theme in teacher education. The purpose of this reported study was to understand the role of guided reflections on mathematical tasks in a college geometry course. We were also interested in understanding how guided reflections on mathematical tasks would effect teachers’ development of MKT. Our research data consist of participants’ reflections, teaching scenarios, and pre-post test results. In this study we developed a workable framework for data analysis. Audience discussion will address questions related to the proposed analysis framework and development of MKT in college mathematics courses.

Effects of Collaborative Revision on Beliefs about Proof Function and Validation Skills

Emily Cilli-Turner

Abstract: Although there is much research showing that proof serves more than just a verification function in mathematics, there is little research documenting which functions of proof undergraduate students understand. Additionally, research suggests that students have difficulty in determining the validity of a given proof. This study examines the effects of a teaching intervention called collaborative revision on student beliefs regarding proof and on student proof validation skills. Student assessment data was collected and interviews were conducted with students in the treatment course and in a comparison course. At the end of this study, we will produce a categorization of the proof functions that students appreciate, as well as a determination of the value of the teaching intervention on students’ abilities to correctly classify proofs as valid or invalid.

Assessing Student Presentations in an Inquiry-Based Learning Course

Christina Eubanks-Turner

Abstract: Inquiry-Based Learning (IBL) is an instruction method that puts the student as the focal point of the learning experience. An integral part of many IBL proof-based courses is presentation of proofs given by students. In this report, I introduce an assessment rubric used to evaluate student presentations of theoretical exercises from a presentation script. The W.I.P.E. rubric is based on several different assessment models, which emphasize proof writing and comprehension. The rubric has been created to evaluate in-class presentations in undergraduate Abstract Algebra courses for math and math education majors, which offer graduate credit.

The Role of Time in a Related Rate Scenario

Katherine Czeranko

Abstract: Students who graduate with an engineering or science degree using applied mathematics are expected to synthesize concepts from calculus to solve problems. First semester calculus students attempting to understand the derivative as a rate of change encounter difficulties. Specifically, the challenges arise while making the decision to apply an average rate of change or an instantaneous rate of change (Zandieh, 2000) to the problem. This paper discusses how students view the derivative in an applied mathematical setting and investigates how the concept of time and other related quantities contribute to the development of a solution.

Calculus Students’ Understanding of Volume

Allison Dorko and Natasha Speer

Abstract: Understanding the concept of volume is important to the learning/understanding of various topics in undergraduate mathematics. Researchers have documented difficulties that elementary school students have in understanding volume, but we know little about college students’ understanding of this topic. This study investigated calculus students’ volume understanding. Clinical interview transcripts and written responses to volume problems were analyzed. Findings include: (1) some calculus students find surface area when directed to find volume, believing that the surface area computation accounts for the object’s three-dimensional space; and (2) some calculus students find volume using reasoning and formulae that contain surface area and volume elements. Comparisons with research on elementary school students’ thinking and implications for the teaching and learning of volume-dependent topics from calculus are also presented.