PROCEEDINGS OF THE 16TH ANNUAL CONFERENCE ON RESEARCH IN UNDERGRADUATE MATHEMATICS EDUCATION (Vol 1)
2013
Denver, Colorado
Number of pages: 422
Number of papers: 31
- STACY BROWN
- GULDEN KARAKOK
- KYEONG HAH ROH
- MICHAEL OEHRTMAN
2013
Denver, Colorado
Number of pages: 422
Number of papers: 31
2013
Denver, Colorado
The purpose of this study is to characterize students’ conceptions of span and linear (in)dependence and to utilize mathematical activities to provide insight into these conceptions. The data under consideration are portions of individual interviews with linear algebra students. Grounded analysis revealed a wide range of student conceptions of span and 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 generating, and problem solving. This framework proves useful in providing finer-grained analyses of students’ conceptions and the potential value and/or limitations of such conceptions in certain contexts.
2013
Denver, Colorado
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 four themes that emerged from the interviews are presented. Educational implications of the results are also discussed.
2013
Denver, Colorado
In this paper, we model students’ concept images and meanings for average and for two- and three-dimensional average rate of change. We use these characterizations to describe how students use their meanings for average to interpret and reason about rate of change. We describe the importance of everyday meanings for average in students’ conceptions of rate, and propose how instruction and activities might address this link. We conclude by discussing the significance of this work for mathematics education, and propose important directions for future research focus on students developing the meanings that instructors intend.
2013
Denver, Colorado
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 𝛿 and 𝜀 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.
2013
Denver, Colorado
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, emphasizing the variety of productive and reflective ways in which the computational mode of thinking is used.
2013
Denver, Colorado
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.
2013
Denver, Colorado
In this paper we describe the results obtained from a diagnostic instrument that was applied to 25 students of the Universidad Autónoma de la Ciudad de México (UACM) at the beginning of an Algebra and Analytic Geometry course. The diagnostic instrument was the first part of the study we currently perform to analyze the possible impact that failing to understand the uses of variables may have in understanding systems of linear equations. A brief description of the main research project is presented to frame the results of the diagnostic instrument into context.
2013
Denver, Colorado
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 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.
2013
Denver, Colorado
This paper describes preliminary results from a larger 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. Namely, as students progress from novice to expert provers, they tend to change proof methods more frequently until they reach the final stages of development (Hart 1994).
2013
Denver, Colorado
In this research, we provide empirical evidence that students can be engaged in theoretical thinking in a university introductory level Calculus course, despite the institutional constraints that often surround and pervade these courses. Students enrolled in a Calculus course were presented with optional tasks intended to engage them in theoretical thinking. We analyze the results from the perspective of Sierpinska et al.’s (2002) model for theoretical thinking; our analysis shows that students who participated in these optional tasks often engaged in theoretical thinking. We discuss these findings in the context of previous research in the teaching and learning of university introductory (and remedial) level mathematics and of the role that Calculus courses play in the mathematics education of undergraduate students.
2013
Denver, Colorado
Completing undergraduate mathematics courses is a central feature of the professional preparation of most prospective secondary mathematics teachers. However, 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 uses a social cognitive perspective to better understand mathematics majors' performance and persistence in undergraduate mathematics courses. Quantitative analysis is accompanied by case studies gleaned from exploratory qualitative 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 secondary mathematics majors in undergraduate mathematics coursework.
2013
Denver, Colorado
This research focuses on mental challenges that students face and how they resolve these challenges while transitioning 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 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 about their task from one form of representation became a model for reasoning into another form of representation within a task. Also, their ways of reasoning from their first task served as referential tools to reason about their next task. In this paper we detail those conceptions and their reasoning from their first task that became model for reasoning within and across subsequent tasks.
2013
Denver, Colorado
Realistic Mathematics Instruction supports students’ formalization of their mathematical activity through guided reinvention. To operationalize “formalization” in a proof-oriented instructional context, I adapt Sjogren’s (2010) claim that formal proof is an explication (Carnap, 1950) of informal proof. Explication is the process of replacing unscientific concepts with scientific ones. I use Carnap’s criteria for successful explication to demonstrate how each element of mathematical theory (definitions, axioms, theorems, proofs) explicates its less formal correlate. I provide examples of students’ proving activity in an axiomatic geometry course to motivate the need for explication, meaning that students should see formal theory as a precise expression of their less formal understandings. I conjecture that students who understand formal theory as an explication will better coordinate semantic and syntactic reasoning toward proof. I provide supporting evidence for this claim from a teaching experiment in which students reinvented axioms and definitions of planar geometry.
2013
Denver, Colorado
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 as described by Black & William’s 2009 framework. This paper examines case studies of two students: Leonard and Sandra. Although the two students earned similar grades, their varying levels of participation in the formative assessments led to very different conceptual development paths in their introductory calculus course.
2013
Denver, Colorado
Researchers have documented difficulties that elementary school students have in understanding volume. Despite its importance in higher mathematics, we know little about college students’ understanding of volume. This study investigated calculus students’ understanding of volume. Clinical interview transcripts and written responses to volume problems were analyzed. One finding is that some calculus students, when asked to find volume, find surface area instead and others blend volume and surface area ideas. We categorize students’ formulae according to their volume and surface area elements. We found that some of these students believe adding the areas of an object’s faces measures three-dimensional space. Findings from interviews also revealed that understanding volume as an array of cubes is connected to successfully solving volume problems. This finding and others are compared to what has been documented for elementary school studens. Implications for calculus teaching and learning are discussed.
2013
Denver, Colorado
An introductory proofs course was taught using Inquiry-based learning (IBL), which gives authority to students and allows them to present to their peers rather than having the instructor as the focus of the class. Data was collected from the final exams of 68 students and analyzed based on proof structure. Problems included concepts that were introduced prior to and during the course. This research utilizes an adaptation of Toulmin’s method for argumentation analysis. Our goal was to compare the proof structures from these students to previous research that applied Toulmin’s layout to mathematical proof. There was a much wider variety of proof structures than expected, which could be a result of the IBL atmosphere.
2013
Denver, Colorado
Students’ understanding of derivative and their difficulties in solving applied problems have been the subject of rich research work. However little research has examined students’ ways of thinking about derivative through the lens of their work on applied questions. The focus of this research is on whether relationships exist between students’ ways of thinking about derivative and success on applied derivative problems. Survey data were used to look at students’ multiple ways of thinking and their work on applied derivative problems. “Multiple ways of thinking” refers to two or more ways of thinking about derivative (e.g., slope of the tangent line at a point on a function or instantaneous rate of change). Findings indicate that students who demonstrate two or more ways of thinking about derivative were able to complete more steps of the applied problems.
2013
Denver, Colorado
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 pedagogies (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 interaction and discourse play in facilitating learning, improving conceptual understanding, and empowering students to engage in future self-initiated communal learning. The goal was to develop an effective classroom model that cultivates advancement in knowledge and enculturation into the STEM community, culminating in a higher retention rate of diverse students in STEM disciplines. Results indicate that a hybrid model encompassing both traditional instruction and cooperative learning successfully enhances students’ self-efficacy.
2013
Denver, Colorado
Like several other research groups, we have been investigating measures for capturing change in middle and high school teachers’ mathematical pedagogical content knowledge (PCK). This report focuses on 14 teachers who have completed a distance-delivered master’s degree in mathematics education. The group is the first of five cohorts who will complete a program that seeks to develop content proficiency, intercultural competence, and pedagogical expertise for teaching mathematics. Analysis included pre- and post-program data from observations of participants at work and written PCK assessments. Results indicate significant changes in curricular content knowledge and discourse knowledge. Path analyses suggest teacher discourse knowledge as measured by the written assessments is significantly related to discourse knowledge as measured by the post-program observation.
2013
Denver, Colorado
Self-inquiry is the process of posing questions to oneself while solving a problem. The self- inquiry of thirteen undergraduate mathematics majors 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. Data 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, questions have arisen about the further exploration of self-inquiry.
2013
Denver, Colorado
This case study explored how a student could use Venn diagrams to explain his reasoning while solving counting problems. Open coding was used to identify the representations he used, and the ways of thinking in which he engaged were analyzed using an existing framework. Venn diagrams were first introduced as part of an alternate solution written by a supposed prior student. Following this introduction, the student in this study often chose to use Venn diagrams to explain his reasoning, stating that he was envisioning them. They were a powerful model for him – they helped him visualize the sets of elements he was counting and to recognize over counting. Further, he adopted the representation of the universal set in his diagrams when posing new questions and finding additive relationships between the solutions of the new and original questions. He transferred this representation to find multiplicative relationships in other problem posing situations.
2013
Denver, Colorado
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 differed in design and in the type of reasoning students evaluate, but they both provoked students to further develop their reasoning, and therefore their understanding. We discuss the implications of these interventions for both research and teaching practice.
2013
Denver, Colorado
This paper offers a theoretical perspective for students’ understanding of sampling, samples, and sampling distributions by melding aspects of the Action, Process, Object, and Schema theory and Saldanha’s and Thompson’s Multiplicative Conception of Sampling. This theoretical perspective provides one potential way to describe the development of a student’s conception of sampling. Additionally this perspective differs from most other perspectives in that it does not focus on the sample size the student uses or the sampling method, but rather how the student understands sampling in terms of a sampling distribution.
2013
Denver, Colorado
The accepted framing of mathematics pedagogical content knowledge (PCK) as mathematical knowledge for teaching has centered on the question: What mathematical reasoning, insight, understanding, and skills are required for a person to teach elementary mathematics? Many have worked to address this question in K-8 teaching. Yet, there remains a call for examples and theory in the context of teachers with greater mathematical preparation and older students with varied and complex experiences in learning mathematics. In this theory development report we offer background and examples for an extended theory of PCK – as the interplay among conceptually-rich mathematical understandings, experience in and of teaching, and multiple culturally-mediated classroom interactions.
2013
Denver, Colorado
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 strategy of solving smaller, similar problems is tied to students’ facility with sets of outcomes. Drawing upon student data from semi-structured 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 combinatorics, and avenues for further research are discussed.
2013
Denver, Colorado
This study is an investigation of the questions that are asked by four faculty members who were teaching advanced mathematics. Observations of each classroom were conducted, and the questions asked by the instructor were analyzed along two dimensions: the expected response type of the question and the Bloom’s Taxonomy level.
2013
Denver, Colorado
Researchers continue to emphasize the importance of covariational reasoning in the context of students’ function concept, particularly when graphing in the Cartesian coordinate system (CCS). In this manuscript, we extend this body of literature by characterizing two pre-service teachers’ thinking during a teaching experiment focused on graphing in the polar coordinate system (PCS). We illustrate how the participants engaged in covariational reasoning to make sense of graphing in the PCS and make connections with graphing in the CCS. By foregrounding covariational relationships, the students came to understand graphs in different coordinate systems as representative of the same relationship despite differences in the perceptual features of these graphs. In synthesizing the students’ activity, we provide remarks on instructional approaches to graphing and how the PCS forms a potential context for promoting covariational reasoning.
2013
Denver, Colorado
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 that explored pre-service teachers’ developing understandings of the PCS. We specifically identify several issues that arose spontaneously as we worked with the students. For instance, we illustrate how students’ angle measure meanings influenced their construction of the PCS. We also discuss ways in which the students’ understandings of the Cartesian coordinate system (CCS) became problematic as they transitioned to the PCS. As an example, mathematical differences between the polar pole and Cartesian origin perturbed some students. Collectively, our findings highlight the emergent nature of students’ coordinate systems.
2013
Denver, Colorado
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 previous study of proving (Savic, 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 (2005) 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 that of the graduate student. This study may influence a proving-process framework that can encompass both the problem- solving aspects of proving and the differences found.
2013
Denver, Colorado
This study discusses various theoretical perspectives on abstract concept formation. Students’ reasoning about abstract objects is described based on a proposition that abstraction is a shift from abstract to concrete. Existing literature suggested a theoretical framework for the study. The framework describes a 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, and 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 a students’ reasoning about abstract objects. The assumptions, therefore, provide implications for instructions and future research.
2013
Denver, Colorado
This paper describes two first-semester calculus students’ meanings for rate of change, and how these meanings shaped their conversations and interpretations of rate in three dimensions. I present the theoretical structure of the teaching experiment in which the two students participated, use excerpts from the teaching experiment to illustrate the development of their ways of understanding, and present a retrospective analysis that characterizes the meanings I believed the students possessed. I conclude by discussing the need to use results from literature about generalization and abstraction to inform the study of student thinking in multivariable calculus.