PROCEEDINGS OF THE 18TH ANNUAL CONFERENCE ON RESEARCH IN UNDERGRADUATE MATHEMATICS EDUCATION
2015
Pittsburgh, Pennsylvania
Number of pages: 1100
Number of papers: 163
- Tim Fukawa-Connelly
- Nicole Engelke Infante
- Karen Keene
- Michelle Zandieh
2015
Pittsburgh, Pennsylvania
Number of pages: 1100
Number of papers: 163
2015
Pittsburgh, Pennsylvania
Motivated by the observation that formal logic answers questions students have not yet asked, we conducted an exploratory teaching experiment with undergraduate students intended to guide their reinvention of truth-functional definitions for basic logical connectives. We intend to bridge the gap between reasoning and logic by inviting students to ask and answer questions that motivate logic as an objective science. We present categories of student strategies for assessing truth-values for mathematical disjunctions. Students’ reasoning heavily reflected content-specific and pragmatic factors in ways inconsistent with the norms and conventions of mathematical, formalized logic. Despite this, all student groups reinvented the standard truth-functional definition for non-quantified disjunctions once they began reasoning about logic by attending to logical connectives and by comparing their interpretations across various disjunctions. Students struggled to develop generalizable tools for assessing quantified disjunctions because they explored sets of examples in context-dependent ways.
2015
Pittsburgh, Pennsylvania
In this paper we examine a commonly suggested proof construction strategy from the mathematics education literature—that students first produce an informal argument and then work to construct a formal proof based on that informal argument. The work of students who produce such informal arguments when solving proof construction tasks was analyzed to distill three activities that contribute to students’ successful translation of informal arguments into formal proofs. These activities are elaborating, syntactifying, and rewarranting. We analyze how engaging in these activities relates to students success in proof construction tasks. Additionally, we discuss how each individual activity contributes to the translation of an informal argument into a formal proof.
2015
Pittsburgh, Pennsylvania
Though commonly used in math and physics, the concept of frames of reference is not described cognitively in any literature. The lack of a careful description of the mental actions involved in thinking within a frame of reference inhibits our ability to account for issues related to frames of reference in students’ reasoning. In this paper we offer a theoretical model of mental actions involved in conceptualizing a frame of reference. Additionally, we posit mental actions that are necessary for a student to reason with multiple frames of reference. This theoretical model provides an additional lens through which researchers can examine students’ quantitative reasoning.
2015
Pittsburgh, Pennsylvania
This theoretical paper suggests a perspective for understanding university students’ proof construction. It is based on the ideas of conceptual and procedural knowledge, explicit and implicit learning, cognitive feelings and beliefs, behavioral schemas, automaticity, working memory, consciousness, and System 1 and System 2 cognition. In particular, we discuss proving actions, such as the construction of proof frameworks that could be automated, thereby reducing the burden on working memory and enabling university students to devote more resources to the truly hard parts of proofs.
2015
Pittsburgh, Pennsylvania
Research on equity in mathematics education has been one of the primary foci among K-12 researchers. However, in research on undergraduate mathematics education, equity research has yet to establish and maintain the presence and consistency that aligns with issues of inequity related to fairness, access and opportunity. In K-12 research, the focus has shifted from individual context to socio-cultural context and now to understanding the social and political aspects of power and identity in mathematics education research. In RUME, the use of theories related to socio-cultural contexts has been increasing but there has not been a major shift in the focus of the research toward addressing issues of equity. In this paper, Gutiérrez’ (2009) four dimensions of equity: access, achievement, power and identity are used as a conceptual framework to interpret insights from K-12 equity research and apply them to existing studies in RUME. The report ends with open questions and directions for equity research at the postsecondary level.
2015
Pittsburgh, Pennsylvania
Pedagogical reforms in undergraduate STEM courses are garnering increasing attention in the literature and from national organizations in disciplines such as mathematics, physics, and chemistry. While there is significant evidence to support the effectiveness of classroom-based pedagogical reforms, the way in which these reforms are taken up by those not involved in their development varies widely. This comparative case study seeks to better understand the ways in which student-centered instructional reforms in undergraduate mathematics are implemented. Data is taken from videotaped instruction of three participating instructors at three different institutions as they work to implement a student-centered instructional unit focused on supporting students’ understanding of span and linear independence in undergraduate linear algebra. This analysis examines the way in which these instructors structured class time when implementing the unit, and the nature of opportunities for students to explain their thinking in the context of whole class discussions.
2015
Pittsburgh, Pennsylvania
The inverted, or flipped, classroom model is attracting the attention of many researchers, practitioners, and administrators in undergraduate mathematics programs as a way to navigate the tension between coverage and engagement, and to respond to the problem of increased class sizes and decreased budgets. The literature contains many reports on successful implementations, which vary widely in content delivery and student engagement. However, a core set of commonalities shared by these successful implementations forms the nucleus of a list of best practices for flipping a class. I discuss the theoretical underpinnings of the inverted model and the best practices suggested by the literature, and examine as a case study an inverted calculus class that did not follow these emerging best practices.
2015
Pittsburgh, Pennsylvania
Proof is a complex mathematical activity with which students struggle as they transition from K- 14 to abstract mathematics. This transition may be eased by developing mathematical practices during more accessible mathematical activities, such as conjecturing. Recent studies in both proof and conjecturing suggest that instantiation practices, practices surrounding the generation and selection of examples during mathematical inquiry, are key to success in both activities. In our own study, most participants utilized GeoGebra, a dynamic geometry software, to facilitate their investigations; however, although this software (in theory) makes sophisticated instantiation practices possible, it did not appear to advance participants’ instantiation practices. In this paper, we detail the participants’ use of the GeoGebra and raise implications, questions, and cautions for teaching and research.
2015
Pittsburgh, Pennsylvania
Evaluating final exams can give insight into what aspects of a course instructors value the most. This study examines whether final exams in proof-based courses accurately reflect instructors’ perceptions of their exams. It also categorizes the questions on the exams in terms of type, format, and cognition level. The level of cognition was further distinguished by imitative reasoning and creative reasoning. Results indicate that instructors of proof- based classes are relatively aware of the content of their exams. The largest discrepancy between instructor perception and reality concerned the number of application problems. Analysis also showed that about a fifth of the questions and over half of the points are proof- related. The exams were almost entirely short or broad answer and required high cognition levels when proofs were assumed to require creativity.
2015
Pittsburgh, Pennsylvania
This preliminary study explores pre-service teachers’ learning and understanding of arithmetic as they are confronted with numbers represented in different bases in order to add to the body of literature about teacher knowledge. Students in an initial mathematics content course for elementary teachers were interviewed as they solved problems related to representing whole numbers and related to arithmetic operations on numbers represented in various bases. Initial findings suggested that the areas of knowledge development during the course were still domain specific and the links, or concepts, in terms of semantic networks were still weakly connected. Students could hold contradictory views of concepts. Additionally, standard algorithms for operations in other bases were also not consolidated; they were not abstracted to the point that new structures had emerged as conceptualized.
2015
Pittsburgh, Pennsylvania
Concept inventories have become increasingly common in STEM disciplines as a means of assessing student conceptual understanding on a given topic, and overall they have led to significant reform in the teaching and learning of content in their respective disciplines. In mathematics, the use of the Calculus Concept Inventory seems, anecdotally and based on a review of the literature, to be growing. Yet peer-reviewed literature on its development and psychometric properties is lacking. Using data from approximately 1800 students across four institutions, we analyzed its content validity, internal structure validity, and reliability. We conclude that the data is consistent with a unidimensional model and that the instrument lacks sufficiently strong reliability for its intended use. Based on these findings, we argue the need for creating and validating a criterion-referenced concept inventory on differential calculus and outline a potential framework for such an instrument.
2015
Pittsburgh, Pennsylvania
This study investigates the ways that undergraduate students use examples in their transition to proof course, and the influence that the instructor had on the students’ decisions to use examples. Data was collected from the instructor and a sample of students via observations and interviews to develop an model of effective example use in a transition-to-proof course. The results show that the students can often state the circumstances in which an example could provide insight during proof writing, but struggle during the implementation.
2015
Pittsburgh, Pennsylvania
Fluency with our number system is a critical part of mathematics. Understanding how rational and irrational numbers work and fit in to the number system as a whole is at the foundation of a good understanding of mathematics. In this study, we present developmental mathematics students with a task which tests understanding of the closure of irrational numbers under addition and multiplication. We analyze the data using the strands of proficiency framework from Adding It Up (Kilpatrick, Swafford, & Findell, 2001), searching for evidence of each strand. The results show varied levels of proficiency within each strand and overall. We conclude with implications for teaching.
2015
Pittsburgh, Pennsylvania
What is happening for in-service teachers at the classroom intersection of mathematics, culture(s), teaching, and learning? How can knowing the answer to that question inform teacher preparation, induction, and development? In ongoing efforts to model and measure the intercultural and relational aspects of pedagogical content knowledge, we present a model and data analyses. The focus is teacher learning and intercultural orientation development. Data are pre- and post-program written tests, surveys, and classroom observations among four cohorts (70 in-service teachers) enrolled in a two-year master’s program. The focus at the conference was harvesting the intellectual power of the audience to consider questions about the connections – qualitative, quantitative, and otherwise – among core constructs in pedagogical content knowledge, the thinking that teachers do in connecting them, and how knowing about intercultural orientation and how it plays out in the classroom can inform teacher education and professional development.
2015
Pittsburgh, Pennsylvania
Using Tall and Vinner’s notion of Concept Image (1981) we analyzed the concepts students used while working with the differential equation 𝑃′=3𝑃 and the connections between these concepts. We identified five interconnected elements in the students image of 𝑃′ in the context of differential equations that played an integral role in the students’ reasoning while attempting to solve interview tasks. In this study we report our findings concerning a pair of students’ beginning to treat a differential equation as a function of two variables by reasoning with their notions of slope and rate. Specifically we address the students’ transition from seeing a differential equation as solely an algorithm for verifying a function is a solution to a differential equation, to treating the differential equation as a relationship between a function’s rate of change and evaluated value.
2015
Pittsburgh, Pennsylvania
Despite its perceived importance in the learning of mathematics, there is a dearth of research on students’ use of mathematical language, particularly when writing proofs at the undergraduate level. In this exploratory study, we analyze written student exams from an introductory proof course and identify fourteen categories of unconventional uses of mathematical language found in undergraduate student generated proofs.
2015
Pittsburgh, Pennsylvania
This paper explores the role examples play as mathematicians formulate conjectures. Although previous research has examined example-related activity during the act of proving, less is known about how examples arise during the formulation of conjectures. We interviewed thirteen mathematicians as they explored tasks requiring the development of conjectures. During the interviews, mathematicians productively used examples as they formulated conjectures, particularly by creating systematic lists of examples that they examined for patterns, an activity that we call “Exhaustive Example Generation.” The results suggest further research and pedagogical implications for explicitly targeting examples in conjecturing, and the study contributes to a body of literature that points to the benefits of exploring, identifying, and leveraging examples in proof-related activity.
2015
Pittsburgh, Pennsylvania
In 2000, the MAA composed a the Curriculum Reform Across the First Two Years report that outlined their recommendations for Business Mathematics, among other mathematical topics. For Business Mathematics, they recommended an emphasis on applications, modeling, and communication of mathematical results. Because of time constraints, adding in more applications and modeling must require cutting out some topics that are traditionally covered in such a Calculus course. This study aims to determine the specific Calculus content that is used in undergraduate level business courses to inform decisions about which content to cover in Business Calculus. This study uses an online survey instrument and interviews to explore what Calculus topics business faculty view as relevant to, and necessary for, various business specializations.
2015
Pittsburgh, Pennsylvania
Many stakeholders concur that secondary teacher preparation programs should include study of abstract algebraic structures, and most certification programs require an abstract algebra course for prospective mathematics teachers. However, research has shown that undergraduate students learning abstract algebra for the first time often struggle to relate the new abstract concepts to previously learned concepts found in secondary school mathematics. In this study, I examined nine undergraduate abstract algebra textbooks to elaborate on the mathematical connections explicitly stated in the textbooks between abstract algebra and secondary school mathematics concepts, and categorized them according to an analytic framework based on categories of mathematical connections found in previous literature. In this report, I discuss three specific categories of connections: comparison through common features, generalization, and hierarchical or inclusion. I found that the connection types discussed in textbooks differ from past literature that highlight connections related to abstract algebra as generalization of school algebra.
2015
Pittsburgh, Pennsylvania
In this study, we examined how fifteen prospective secondary mathematics teachers validated five different arguments purported to be proofs that the sum of the first n odd natural numbers is n2 in an interview setting. We also investigated the participants’ stated expectations for middle student proof work. Our participants had varying evaluation criteria when validating arguments, but largely focused on generality and proof form or appearance. The majority of our participants expected only the empirical argument to be given by a middle school student. This was also the argument fewest participants accepted as a proof. In general, our participants were less likely to expect an argument from middle school students if they accepted it as a proof. Further research is needed to examine how argumentation experiences during their teacher preparation program influence prospective teachers’ expectations of students’ abilities to produce appropriate arguments.
2015
Pittsburgh, Pennsylvania
Studies have shown that students have difficulty with the concepts of slope and derivative, especially in the case of real-life contexts. Following up from a previous study, written surveys were used to collect data from 69 differential calculus students, and 7 clinical interviews were conducted. On the surveys, students answered questions about linear and nonlinear relationships and interpretations of slope and derivative. They also critiqued the reasoning and accuracy of a hypothetical person’s predictions. In interviews, students explained their thought processes and reasoning while answering questions similar to the written survey but in a different context. Results indicate that students struggle with knowing what the derivative represents and how to use it appropriately to make predictions. These struggles might stem in part from students’ incorrect interpretations of slope as a ratio of totals, as opposed to a ratio of changes over a particular interval.
2015
Pittsburgh, Pennsylvania
This study explored the effect of participation in Math Teachers’ Circles (MTC) on aspects of teacher leadership. MTCs are a collaborative professional development model, aimed primarily at middle school mathematics teachers, focused on problem-solving and open-ended mathematical explorations. Prior studies have linked them to increased content knowledge and pedagogical content knowledge, particularly in the domains of number and operation. This exploratory study used self-report data gathered from a national survey across 13 MTC sites and consisting of 169 completed surveys from MTC participants. Analysis of the data showed that MTC participation can help teachers develop across the three stages of leadership presented in the PRIME leadership framework. Further research is needed to ascertain characteristics of teachers for whom this effect is particularly pronounced, as well as to reconcile some of the self- report data with actual practice.
2015
Pittsburgh, Pennsylvania
This paper offers a model of the structure of proof construction. The model provides a comprehensive view of proof construction, which can encompass the aspects, factors, patterns, and features involved in cognitive processes in proof construction. The model also clarifies the skills and abilities necessary for proof construction. Moreover, the model provides an algorithm for advancing a reasoning process. Using some examples, this paper shows that the model can serve as metacognitive and methodological knowledge to help students construct a proof based on logical deduction. The model may help students not only grasp a view of proof construction but also develop their skills for proof construction.
2015
Pittsburgh, Pennsylvania
This paper examines students’ difficulties with starting a proof. The target population was those students who were enrolled in undergraduate Algebra, Analysis, and Topology in a large research university. 81 proofs, which were collected from students’ mid-term and final exams and in-class problem solving sessions, were analyzed. The results showed that 39 proofs (about 48%) were not successful because the early stages of those proofs had defects. This paper investigates the sources of students’ difficulties with starting a proof and provides pedagogical suggestions to help them overcome their difficulties.
2015
Pittsburgh, Pennsylvania
Proof is a central and essential skill in mathematics. However, it is a challenging task for students at all levels. This paper presents the findings from the analysis of students’ difficulties with proof construction, clarifies possible sources of their difficulties in light of the structure of proof construction, and offers an algorithm for proof construction as metacognitive knowledge for helping students with proof construction based on logical deduction.
2015
Pittsburgh, Pennsylvania
This study explores teachers’ meanings for average rate of change in U.S.A. and Korea. We believe that teachers convey their meanings to students and teachers who have productive mathematical meanings help students build coherent meanings. We administered a diagnostic instrument to 96 U.S. teachers and 66 Korean teachers. Some of teachers’ responses revealed particular problematic meanings for average rate of change that should be addressed in professional development. Our analyses suggest that Korean teachers’ meanings for average rate of change are substantially stronger than U.S. teachers’ meanings.
2015
Pittsburgh, Pennsylvania
2015
Pittsburgh, Pennsylvania
The area of focus for the 2015 Mathematics & Physics Working Group was the interface of student understanding of linear algebra and its application in physics courses, such as Quantum Mechanics. This working group brought together researchers in Undergraduate Mathematics Education and Physics Education to analyze data, compare methods for analysis, and highlight areas of interest or focus from within the data on student thinking. This focus was intended to foster discussions and possible collaborations about research at the interface of RUME and PER, in linear algebra and beyond, broadly conceived.
2015
Pittsburgh, Pennsylvania
The Research on College Mathematics Instructor Professional Development working group meets online several times per year and annually at the RUME conference. We solicit proposals from researchers in all areas of the professional development of college mathematics instructors across institutional types (e.g., community college, university). This includes, but is not limited to, research on factors that shape instructional practices and the experiences of instructors as they attend to student thinking in their instruction. The group’s goals, historically and as they have evolved, continue to drive the focus of annual meetings. They include interaction that offers (1) informed support and feedback for researchers, (2) opportunities for networking and collaboration among mathematics educators interested in research and development of materials, processes, models, and theories to support the professional development of collegiate mathematics instructors, (3) continuing discussion of issues central to the field and ways to address them. The intended participants of this group include researchers in all of these areas, whether new to the field, to research in general (early career researchers) or experienced. Researchers need not present their work to participate in the group or provide feedback to others. Group meeting time is structured to allow feedback on research projects that are in progress. The working group is not meant to be a forum for presenting completed studies, but rather an opportunity to get feedback from peers on projects in any stage; from the refinement of research questions to study design, data collection and analysis, to discussion of venues for future presentation and proposals for funding of projects. We also discuss strategies for sharing our work with the practice-oriented college mathematics instructor professional development community, the needs of the working group, and ways of sustaining collaborations and communication among group participants during the year.
2015
Pittsburgh, Pennsylvania
University students often encounter difficulties making correct use of definitions. This is partly because they are not familiar with the difference between dictionary definitions and mathematical definitions (Edwards & Ward, 2004). However, in order to succeed in upper-level mathematics courses, they must often construct original (to them) mathematical proofs, which intrinsically require the correct use of definitions. Only a little research has been conducted to discover how university students handle definitions new to them (e.g., Dahlberg & Housman, 1997). Our research question is: How do university students use definitions, to evaluate and justify examples and non-examples, to prove results, and to evaluate and justify true/false statements? Data were collected through individual task-based interviews with 23 volunteer students from a transition-to- proof course. There were five definitions but each student considered only one of the five. Content analysis and grounded theory are being used for analysis. Preliminary results are presented.
2015
Pittsburgh, Pennsylvania
Calculus is typically the first undergraduate mathematics course for science, technology, engineering, and mathematics (STEM) majors in the United States. Internationally as well as domestically, first year mathematics courses are credited with preventing students from continuing along STEM paths. A recent study of the features that characterize exemplary calculus programs from five PhD-granting institutions highlighted several common characteristics, one of which was the existence of a well-established system for coordinating Calculus I. This coordination of courses and instructors seems to engender a community of practice. This study aims to expand on this finding by leveraging social network theory to map the underlying structure of the social ties between instructors of lower-division undergraduate mathematics courses, to compare informal and formal organizational structures in each case, and to compare the communities across the five selected institutions. Here I report on the results from one of the five selected institutions.
2015
Pittsburgh, Pennsylvania
This poster will explore the various representations of groups found within introductory abstract algebra textbooks. Representations play an essential role in students understanding of mathematics (Goldin, 2002). Textbooks provide one source for analyzing the intended curriculum and what representations students may have access to within their introductory course.
2015
Pittsburgh, Pennsylvania
De Villiers (1990) suggested five roles of proof important in the professional mathematics community that may also serve to meaningfully engage students in learning proof: verification, explanation, systematization, discovery, and communication. We investigate written reflections on an end-of-semester assignment from undergraduates in an inquiry-based transition to proof course, where students reflected on instances during the semester when they engaged in the five roles of proof. We present (a) student rankings of role engagement, (b) the types of activities students recalled as influential to their engagement in the roles of proof, and (c) how students perceived they engaged in the five roles. Students in this course reflected on activities distinctive of the inquiry-based environment (such as discussing, presenting, conjecturing, and critiquing) as influential to their engagement in the roles of proof. We provide student quotations highlighting these activities and offer implications for both research and practice.
2015
Pittsburgh, Pennsylvania
The national goal of increasing STEM majors is well-established. As noted in “Rising Above the Gathering Storm” (National Academies, 2008); “Preparing the Next Generation of STEM Innovators: Identifying and Developing Our Nation’s Human” Capital (National Science Board, 2010) and several other recent publications (National Academy of Science, 2011; CRS Report, 2008; National Academy of Science, 2010; National Academy of Science, 2012), it is widely recognized that sustaining the economic leadership of the United States over the next two decades will require a robust supply of bachelor’s degree graduates in the STEM disciplines. Certain specific courses within the first two years in mathematics and the sciences are rate-limiting for students that have a realistic chance of success in a major in science, technology, engineering, or mathematics (STEM), but “bog down” at a particular point in their trajectory of courses. Identification of such points, and application of resources to improve student success at those points, can dramatically improve retention in STEM.
2015
Pittsburgh, Pennsylvania
Developmental mathematics courses in the United States continue to lack in curriculum and instructional practices that promote students’ mathematical proficiency. The instructional practices that researchers argue can promote students’ mathematical proficiency in K-12 classrooms can apply to undergraduate classrooms as well. This poster will discuss an NSF-funded research project that facilitates students’ mathematical development in a non-credit-bearing developmental mathematics course, in concert with providing prospective mathematics teachers an opportunity to learn to teach for mathematical proficiency. The project team collected quantitative data that compared the intervention students to their peers before and after participation in the course. Results indicated that developmental mathematics students who participated in our intervention started behind, caught up, and experienced more success than their peers in their subsequent mathematics course.
2015
Pittsburgh, Pennsylvania
The aims of this study are two-fold: (1) to investigate novices’ proof preferences as indicated by novices’ selection of the “most convincing” argument when engaging in proof comparison tasks involving an indirect and a direct proof; and, (2) to explore the criteria students’ bring to bear on proofs as they engage in proof comparisons. Informed by the cK¢ model of conceptions proposed by Balacheff (2010), analyses indicate that directness was not a primary criterion used for the select of a proof during proof comparisons, even though this criterion was suggested by prior research. Instead, the primary criteria identified in students’ rationales were familiarity and the degree of certainty in one’s understanding of the given proofs, which in turn suggests that it is the subjective sense of being on solid conceptual grounds that determine students’ preferences. These findings are considered in light of the cK¢ model of conceptions.
2015
Pittsburgh, Pennsylvania
Although a mathematical proof is a syntactic product, the proving process often entails other reasoning types, such as semantic or intuitive, that contribute to the evaluation of conjectures and the construction of supporting arguments. Cognitive unity and rupture refer to the possible continuity or discontinuity, respectively, between various reasoning types, argumentation and mathematical proof, and the processes of evaluating and proving conjectures. Undergraduate students struggle with mathematical proof, but it is hypothesized that cognitive unity facilitates the proving process. In this study, task-based interviews were conducted with undergraduate students who completed three prove-or-disprove tasks. The goals are to determine conditions under which students experience cognitive unity or rupture when evaluating and proving the conjectures and conditions under which cognitive unity and rupture aided or hindered the proving process. Preliminary findings suggest that various factors affect cognitive unity, cognitive unity can hinder proving, and cognitive rupture can facilitate proving.
2015
Pittsburgh, Pennsylvania
There are many reports on the impact of Supplemental Instruction (SI) on grades and retention of students within STEM disciplines. Although we know that many factors contribute to higher academic success (e.g., increased content knowledge, motivation toward subject, and/or time on task), there is little evidence for the mechanism by which SI improves student performance in any of these factors. This report is a preliminary analysis of an ongoing, multi-disciplinary study, using preexisting valid and reliable content knowledge assessments and attitude surveys, to gain insight into the effects of SI on STEM students.
2015
Pittsburgh, Pennsylvania
In undergraduate mathematics, deductive reasoning is an important skill in the learning of theoretical ideas. Deductive reasoning is primarily characterized by the concept of logical implication (inferring what follows from a given premise). This plays a role whenever mathematical theorems are applied, i.e. one must first check if a theorem’s hypothesis is satisfied and then make a correct inference. In Calculus, students must learn how to apply theorems. However, most undergraduates have not yet received extensive training in propositional logic. How do these students comprehend the notion of logical implication and how does it relate to their understanding of theorems? Results from a pilot study indicated that students struggled with the notion of logical implication in both symbolic and Calculus contexts. However, findings were inconclusive regarding the relationship between the two areas. Background on the current literature, results of the pilot study, and further avenues of inquiry are discussed.
2015
Pittsburgh, Pennsylvania
Within the context of Taylor series expansions as approximations, we illustrate the context dependence of student reasoning about these approximations - specifically the ways in which students’ notions of what is appropriate in mathematics, physics, or engineering, drive how they engage in and reflect on the solutions they produce. Using data from semi- structured interviews, we build on previous work to argue that students' epistemological framing not only plays a role in their choice of solution strategies, but also how they feel those solution strategies would be perceived within various disciplines.
2015
Pittsburgh, Pennsylvania
Despite intuitive foundations, the nature of the transition to abstract topology often results in students’ reliance on dissociated collections of definitions and theorems, without any integrated cognitive structure. In recent decades, there have been numerous analyses of proof, symbols, and the encapsulation of processes as factors in student comprehension, as well as content-specific studies examining which mental constructions support the development of coherent schemata for particular topics. I will expand this research by categorizing students’ understanding in the domain of topology. In a year-long, mixed-methods study, I will analyze the components involved in the development of an axiomatic schema for continuous functions in topological contexts. I will compare this model with actual student constructions in an introductory topology course, collected through task-based interviews and a path-analysis on the coded data. The goal is to confirm the theoretical model, or to provide support for altering the model to increase its validity.
2015
Pittsburgh, Pennsylvania
This research characterizes the cognitive challenges that students encounter while constructing definite integrals to model physical quantities and relationships, how students resolve those challenges, and the resulting conceptual artifacts. We video-recorded four groups of second- semester calculus students working with definite integral models during two 50-minute labs. This paper focuses on how two groups of students that worked on the same problem reasoned about and constructed an integral representing the gravitational attraction between a co-linear thin rod and point mass. Prior research on students’ understanding of definite integrals posits the multiplicative structure f (x)x in a Riemann sum as an essential component in conceiving an integral. Our findings indicate that in many contexts, other symbolic forms subsume the simple product in this essential conceptual role and that they interact significantly with students’ symbolic forms for the definite integral.
2015
Pittsburgh, Pennsylvania
Recent studies have highlighted the positive effects on learning and retention rates that active learning brings to the classroom. A flipped classroom is a type of active learning where transmission of content occurs outside of the classroom environment and problem solving and learning activities become the focus of classroom time. This article reports on results of a study conducted in flipped and non-flipped introductory statistics classroom environments measuring student achievement in both classrooms on traditional assessments as well as measuring student attitudes toward the flipped classroom environment.
2015
Pittsburgh, Pennsylvania
Typical undergraduate course sequences in abstract algebra initiate with group theory before proceeding to ring theory. This sequencing, along with the structural similarities between groups and rings, enables many ring-theoretic concepts to be formulated in terms of results from group theory. What remains to be seen, however, is the extent to which students are able to transfer their knowledge of groups while studying topics in ring theory. Using Wagner's transfer in pieces framework, we conducted an exploratory study to investigate how students in an inquiry-oriented classroom capitalized on their knowledge of groups to make sense of rings. Preliminary results indicate both instances of obvious transfer (e.g. subgroup to subring) and also more creative approaches that might lend insight into how students think about ring structure (e.g. characterizing field-like structures as ‘abelian grouprings’).
2015
Pittsburgh, Pennsylvania
During exploratory teaching experiments intended to guide students to reinvent basic truthfunc- tional definitions for basic logical connectives, it unexpectedly emerged that undergraduate stu- dents reasoned about negation and negative properties in ways incompatible with the conventions and norms of mathematical practice. Specifically, our study participants often unpacked negative properties (not a rectangle) in terms of positive properties (is a parallelogram), which we call semantic negation. In this way, students did not readily adopt the mathematical assumption that the logical negation of a predicate designates the complement of the set of examples that satisfy that predicate. Students especially understood geometric sets of objects as being partitioned by familiar categories rather than those stipulated in a given statement. Semantic negation inhibited students systematizing activity regarding linguistic interpretation because their reasoning about disjunctions in various mathematical contexts depended intrinsically upon their understanding of particular topics so as to preclude abstractions approximating normative logical tools.
2015
Pittsburgh, Pennsylvania
In mathematics, it is a common phenomenon to find several proofs for a single result. This inevitably leads us to believe that proofs are far more than a convincing argument. Indeed, it appears that there is considerable interest in the insight that is gained from the reasoning utilized in a proof. With the existence of several proofs of the same theorem, we are then confronted with a question of value judgment, as it is not necessarily the case that one values all proofs of a given theorem equally. In this theoretical report, I attempt to provide a framework that contributes to the discussion regarding value judgments about proofs by providing a comparative language to systematically talk about judgments one may attach to a proof. I argue that proofs can be valued for reasons such as (1) comprehensibility, (2) explanatory power, (3) originality and surprises, and (4) generalizability.
2015
Pittsburgh, Pennsylvania
In typical undergraduate advanced mathematics courses, professors spend ample class time presenting proofs; however little is know with regards to what students actually gain from these experiences. In our preliminary report we will attempt to address this gap, specifically with respect to a proof of Lagrange’s theorem. We used Mejia-Ramos and colleagues’ (2012) model in designing a proof comprehension test and task-based interviews to shed light on (1) the extent to which undergraduates comprehend the proof and (2) what undergraduates gain or learn from reading the proof. Initial examination of our data reveals that although the participants could follow the proof line by line, they had difficulty identifying key ideas and summarizing the proof. Participants acknowledged their responsibility to fill in gaps in proofs; yet they had trouble justifying non-trivial assertions. Despite participants’ superficial comprehension of the proof, we still observed that participants gained conviction and learned new definitions.
2015
Pittsburgh, Pennsylvania
Most of the attrition from STEM majors occurs between the first two semesters of calculus, and prospective life science majors are one of the groups with the highest attrition rate. One of the largest factors for students that persist in STEM major beyond the first semester of calculus was a sense of community and a perceived connection with their instructor. Since building a sense of community is one of the stated purposes of formative assessment, we investigated to what extent formative assessments could help build a sense of community in a calculus for life science majors course. Two cases of formative assessment used in two sections of this course will be discussed. When implemented as intended, the formative assessments completed weekly by the students made a positive contribution to students' sense of classroom community and their perceived connection with their instructor.
2015
Pittsburgh, Pennsylvania
Generalization is a critical component of mathematical thought and thus a goal of instruction. However, most generalization research has focused on pattern generalization and generalization in algebra, not undergraduate mathematics. It is not known whether the generalization frameworks derived from such work adequately describe generalization in undergraduate mathematics, where the sorts of generalizations students must make are not regarding patterns, but rather the meanings of concepts and ideas. In this paper, we propose elaborations on existing generalization frameworks in order to take into account issues in learning advanced mathematics. We use how students generalize their notion of integration from single to multiple integrals as an illustrative case study.
2015
Pittsburgh, Pennsylvania
The research presented in this paper aimed to construct models of understanding processes of students learning the derivative concepts. The models are constructed following Duval’s theoretical framework and Hitt’s ideas on representations. A teaching experiment was designed to observe students in action while they were participating to teaching episodes on the introduction of the derivative. Preliminary results for one participant are presented.
2015
Pittsburgh, Pennsylvania
This poster describes an intended meaning for the argument of a function when reasoning about a function covariationally. An instructional investigation was designed to promote this meaning with students in a college-level precalculus course using an instructional task and a didactic object relating to modeling the relationship between the time elapsed since Tommy threw a rock off of a bridge in seconds and the height above a lake in feet of the rock. Students were formally introduced to the term argument after the task and asked to evaluate and explain the meaning of functions defined with an argument as the function input. After engaging with the task, these students demonstrated larger learning gains than students in previous studies on the same questions that involved evaluating a function for an argument expression.
2015
Pittsburgh, Pennsylvania
Many undergraduate examinations permit students to use a limited quantity of previously prepared notes: so-called “crib sheets”, or “cheat-sheets”. The majority of evidence from the literature suggests that students sitting such exams feel less anxious, and that they perform to a higher standard, although such results are idiosyncratic to discipline and course, and few are set in the context of undergraduate mathematics. Less is known about what content students choose to include on such a sheet, and how they interact with this material. This preliminary research report presents the first results from a three-year project investigating students’ use of crib sheets in undergraduate mathematics exams. It explores the content and layout of crib sheets used by students for an end-of-semester calculus exam.
2015
Pittsburgh, Pennsylvania
In this study we investigate the relationship between the nature of Calculus I homework and student success. We examine these connections at five PhD granting institutions that were identified in a large US national study as having relatively successful Calculus I programs (compared to similar institutions) and we draw on both qualitative and quantitative from this study. Mixed method analyses point to a clear relationship between homework systems with varied structure, high feedback, and varied content emphases at more successful Calculus I programs where students persist onto Calculus II at higher rates and where students maintain more positive dispositions towards mathematics.
2015
Pittsburgh, Pennsylvania
The goal of this research is to better understand the relationship between how quickly or deeply Calculus material is covered and how this is related to students’ instructional experience and their persistence in a STEM major. To do so, we analyze data coming from a large national survey of Calculus I programs. Specifically, we first compare students’ views of pacing to their instructor’s views, resulting in four classifications of students. We then investigate various characteristics of these students, including: their institution type, their instructor type, their reported instruction, their mathematics beliefs, and their Calculus II intentions. Our findings suggest two important ways that pressure to cover material impacts Calculus I students and their instructors. First, when instructors feel pressure to cover material, student-centered teaching practices are often dropped. Second, feeling rushed to cover difficult material is a factor in losing STEM intending students.
2015
Pittsburgh, Pennsylvania
Research shows that students learn best when working with their peers as this creates opportunities for students to challenge one another’s thinking, consequently building new knowledge. We will share analysis of three groups of calculus students engaging with an activity from a new reform-oriented calculus curriculum, which developed out of the philosophy that students working together promotes increased learning. Inspired by Gresalfi’s engagement framework (2013), we developed an engagement coding scheme that differentiates between weak, procedural, and conceptual engagement. Our analysis focuses on the opportunities that emerged for students to address misconceptions. Findings show that how groups engaged with the same task varied significantly and, although opportunities to address misconceptions emerged in all groups, misconceptions were most effectively addressed in the group where conceptual engagement was prevalent. We discuss the extent to which norms, learning dispositions, and the activity itself played a role in the outcomes we observed.
2015
Pittsburgh, Pennsylvania
Mathematical play, as a mathematical practice, can be defined as the exploration of mathematical ideas in uninhibited and unconstrained ways, which could include engagement with physical devices, computer programs, imaginative acts, social interactions, and inscriptions. In this preliminary report I will discuss mathematical play and the ways in which it transpired in an undergraduate Foundation of Geometry course in which students engaged in activities to develop an understanding of physical, synthetic and analytic aspects of projective geometry. Additionally, I will discuss the ways in which mathematical play can be fostered through engagement in mathematically inspired art projects, as well as other artistic engagement.
2015
Pittsburgh, Pennsylvania
In this poster I describe an instructional sequence for supporting the guided reinvention of multidigit multiplication in a preservice teacher content course. I use design heuristics from Realistic Mathematics Education (Gravemeijer, 1998) to design, test, and modify an instructional sequence leveraging the non-routine context of base five.
2015
Pittsburgh, Pennsylvania
Previous studies have indicated that effective teaching relies on teachers’ knowledge of both student thinking and subject content. Very little is known about the integration (combination) of teachers’ mathematical knowledge and science knowledge for teaching important topics like applied derivative problems. Using Ball and colleagues’ framework for Mathematical Knowledge for Teaching (MKT), data were analyzed from interviews of eight calculus Graduate Teaching Assistants (GTAs) to examine the kind of knowledge used when talking about teaching applied derivative problems. Findings suggest that some of the domains of the existing MKT framework describe the kinds of knowledge GTAs draw on. However, not all elements of knowledge these GTAs used when discussing applied problems fit the MKT framework. Modifications to the framework are proposed to describe teachers’ Integrated Mathematics and Science Knowledge for Teaching.
2015
Pittsburgh, Pennsylvania
Mathematics education research on dynamic technologies incorporated into learning environments indicates that they possess the ability to enrich students’ mathematical conceptual understanding. This study explores how three community college students conceptualize rational functions by classifying their mathematical thinking according to the APOS theory. This study highlighted students’ conceptualization of rational functions as actions (required sequences of observable external steps), processes (sequences with no need to externalize), and objects (thinking about rational functions as a single object or entity).
2015
Pittsburgh, Pennsylvania
Lecture has often been critiqued as obscuring the mathematical habits such as sense-making about abstract statements about mathematical concepts, the creation of conjectures about those concepts, and to the processes of proof-writing and exhibition of counter-examples. In all of these actions, a mathematician or student should be able to draw upon a rich store of examples in order to make meaningful progress and researchers have argued that it is via examples that they are best able to engage in such processes. Inquiry-based classes have, as part of their promise, making more visible the mathematical promises that lecture obscures behind the polished formalism. This preliminary report explores whether and how students in an inquiry- based abstract algebra class engage in public example-based reasoning as a means to explore public versus private mathematizing.
2015
Pittsburgh, Pennsylvania
This paper is a case study of the teaching of an undergraduate abstract algebra course; in particular, it examines the opportunity that the students had during the first class meeting to learn about the concept of a mathematical group and how much intellectual responsibility for the definition they were given. The paper offers a description of the classroom teaching and discussion an inquiry-oriented abstract algebra course. It focuses on the mathematical tasks that the professor used to introduce the concept of a group, the student solutions and classroom presentation of those tasks, and the professor's activities. It analyzes the class in terms of providing students the opportunity to learn the concept of a mathematical group and about the mathematical process of defining. It also analyzes the professor's activities in terms of devolving intellectual responsibility for writing the definition of a group. Finally, the analysis suggests ways that the professor's actions failed to devolve mathematical authority and limited opportunities to learn.
2015
Pittsburgh, Pennsylvania
In this study, we discuss our experience with cooperative learning in a mathematics content course. Twenty undergraduate students from a southern public university participated in this study. The instructional method used in the classroom was cooperative. We rely on previous research and literature to guide the implementation of cooperative learning in the class. The goal of our study is to investigate the relationship between students’ preferences and performance in a cooperative learning setting. We collected data through assessments, surveys, and observations. Results show no significant difference in the comparison of students’ preferences and performance. Based on this study, we provide suggestions in teaching mathematics content courses for prospective teachers in a cooperative learning setting.
2015
Pittsburgh, Pennsylvania
There is a growing body of literature that suggests students do better in classes that implement active-learning strategies (e.g., group work, problem solving). Flipping classroom instruction is becoming a popular innovation to support active learning in the classroom. This preliminary report highlights one instructor’s experiences when implementing ambitious teaching practices in a flipped large lecture Calculus I course. Data collection consists of email correspondence, open-ended interviews, and weekly questionnaires. Particular attention will be made to address how the instructors’ professional identity impacted instruction throughout the term.
2015
Pittsburgh, Pennsylvania
There is a growing body of literature that suggests students do better in classes that implement active-learning strategies (e.g., group work, problem solving). Flipping classroom instruction is becoming a popular innovation to support active learning in the classroom. The study reports on one large lecture calculus course where instruction was flipped in order to employ active-learning strategies. This poster will highlight the experiences of the large lecture instructor, recitation leaders, and student focus groups.
2015
Pittsburgh, Pennsylvania
Policy and standards documents from recent decades emphasize the importance of attending to students’ development of ways of engaging in mathematical activities such as generalizing. Therefore, we need to be able to describe proficient generalizing and ways of explaining and predicting the strategies that individuals use to generalize. However, existing research has largely focused on examining generalizing at the K-12 level on a specific type of task. A broader understanding of mathematical generalizing is important for developing potential developmental trajectories. This report presents tentative findings from the preliminary analysis of data collected as part of the author’s dissertation research on generalizing among university students.
2015
Pittsburgh, Pennsylvania
Two important parts of mathematical proficiency are the individual’s understanding of mathematics and his or her self-efficacy beliefs with respect to mathematics. Understanding the nature of mathematics should include recognizing mathematics as a field of inquiry. However, it is not clear how changes in a student’s understanding of the nature of mathematics might affect the ways that the student perceives his or her mathematical abilities. We designed a 5-week course around inquiry projects in an attempt to promote a more robust understanding of the nature of mathematics. Using surveys and interviews, we gathered information about students’ perceptions of mathematics and about their own mathematical abilities. Our exploratory research suggests that engaging in an inquiry-based course experience helped students to recognize the roles of collaboration and communication in mathematics, but may also have led them to perceive mathematics as inherently more difficult and themselves as less able to communicate mathematically.
2015
Pittsburgh, Pennsylvania
Self-inquiry is the process of posing questions to oneself while solving a problem. The self- inquiry of thirteen undergraduate mathematics students and one mathematics professor was explored. Student self-inquiry was explored via structured interviews requiring the solution of both mathematical and non-mathematical problems. The professor’s self-inquiry was explored through self-reporting of questions asked in an advanced problem-solving context. Using transcripts of the student interviews, a coding scheme for questions posed was developed and all questions were coded. Data analysis of the posed questions suggests that the “good” mathematics students focus more questions on legitimizing their work and fewer questions on specification of the problem-solving task. Data analysis of the professor’s self-inquiry is ongoing and will be compared and contrasted to that of the undergraduates.
2015
Pittsburgh, Pennsylvania
Function is a critical concept for mathematics and related branches of sciences. Despite its centrality in K-12 and undergraduate mathematics, it is one of the most challenging topics to master in school learning due to the various notions associated with the concept (Eisenberg, 1991). Existing research highlights many challenges students face as they learn about functions. Researchers argue that students often associate function with a single algebraic expression or rule (Vinner & Dreyfus, 1989) and have difficulties with piecewise functions (Sfard, 1992). Students also struggle with conceptualizing function both as a process and a mathematical object (Gray & Tall, 1994; Sfard, 1992) and moving flexibly across graphical, algebraic, tabular, and verbal representations of functions (Monk, 1994; Sierpinska, 1992; Tall, 1996). They have difficulties interpreting dynamic representations and information demonstrating the relationship between two variables (Carlson, 1998; Kaput, 1992; Monk, 1994). Given the multiplicity of notions and complexities inherent in the learning of functions, it is important for teachers to address their students’ difficulties in the classrooms. In order to this, teachers need to understand the content and the challenges it presents to students.
2015
Pittsburgh, Pennsylvania
Much of what we know about student understanding of concavity and inflection points comes incidentally from studies looking at the calculus activity of sketching the graphs of functions. However, since concavity and inflection points can be useful in conveying information in disciplines like physics, engineering, biology, and economics, it seems important to study how students understand these two concepts in these contexts. This study attempts to provide insight into this area.
2015
Pittsburgh, Pennsylvania
2015
Pittsburgh, Pennsylvania
Linear algebra is a required course for STEM majors. Many undergraduate students struggle with the sudden exposure to abstraction which is almost an unavoidable feature of the course. Although research on students’ conceptual understanding of linear algebra is going forward, no research has focused on how students react to the order in which the concepts are taught. In this study, we use Tall’s three-world model of mathematical thinking to examine students’ performance in a first year linear algebra course. The study examined two sections of a linear algebra course simultaneously while the instructor changed the order in which she taught the concepts in each class. The result of this investigation so far suggests no significant difference on students’ performance.
2015
Pittsburgh, Pennsylvania
In this report, we draw on the Characteristics of Successful Programs in College Calculus (CSPCC) survey and interview data in order to better understand the student supports available for Calculus I students. We began by investigating possible differences in the availability of supports offered by institutions that were identified as having a successful Calculus I program. This quantitative analysis found no statistical differences in regards to tutoring centers at institutions that were identified as successful and tutoring centers at the other institutions in our study. We then analyzed student interview data in order to investigate how the supports offered at these institutions contribute to student success. This qualitative analysis identified several ways in which academic support systems (and tutoring centers in particular) can facilitate social and academic integration for Calculus I students.
2015
Pittsburgh, Pennsylvania
Undergraduate students need proficient knowledge of the real numbers, because the real numbers form the foundation for more advanced mathematical concepts. To achieve numerical literacy, students must have some proficiency in the real number system (Fischbein, Jahiam, & Cohen, 1995; Guven, Cekmez, & Karatas, 2011). Studies have shown that the set of irrational numbers is difficult to grasp, because of challenges with the definitions of rational and irrational numbers (Fischbein et al., 1995), with the connection between irrational numbers and limits (Peled & Hershkovitz, 1999), and with moving between multiple representations of irrational numbers (Arcavi, Bruckheimer, & Ben-Zvi, 1987; Sirotic & Zazkis, 2007).
2015
Pittsburgh, Pennsylvania
In this poster presentation we bring together different characterizations of mathematical thinking, doing and behaving that researchers have brought forward over the last three decades, to compose, in a way of speaking, a collage! of what we came to call mathematical! behaviors. In previous work, we have combined some of these characterizations to identify opportunities to engage in mathematical behaviors that students may encounter in undergraduate courses, and to design and analyze tasks that may foster the development of such behaviors. The poster format allows us to play with a pictorial representation that reveals the relations and complementarities between the ‘images’ of the collage. We hope to discuss (a) these relations and complementarities, (b) strategies to foster the development of mathematical behaviors in undergraduate mathematics students, and (c) methods that may allow us, as researchers or as teachers, to compose accounts of students’ mathematical behaviors.
2015
Pittsburgh, Pennsylvania
Mathematics educators have long advocated for early algebra to be introduced into the elementary grades. However, little research is currently available to inform teacher preparation programs about the work of preparing undergraduate pre-service teachers for teaching early algebra. The research reported here examines the experiences of undergraduate pre-service teachers as they make consequential transitions from formal to early algebra. Preliminary results suggest that making this transition is far from trivial for undergraduates and that, to varying degrees, they face four kinds of conceptual challenges.
2015
Pittsburgh, Pennsylvania
This poster investigates two students’ use of examples in the transition to advanced mathematical thinking in the context of a teaching experiment focusing on quotient groups. Based upon the theoretical perspective of Edwards, Dubinsky, and McDonald (2008), this poster extends the construct of imperfect models to include instances of generic examples. This study found that students leveraged examples to construct conjectures, form justifications, clarify misconceptions, and provide analogies for further reasoning. However, attending to the specific features of particular examples also led to over-generalizations and invalid reasoning.
2015
Pittsburgh, Pennsylvania
This paper reports on two commognitive conflicts that were identified in a larger research study conducted on university first year students’ discourse on continuous functions. The study looks at continuity related aspects under a participationist, discursive lens in contrast to previous studies on continuity that have used cognitive theories that hold an acquisitionist view on learning. The study adopts Sfard’s commognitive framework to analyze data. Among findings are different ways in which students use the word ‘domain’ and how they struggle with inconsistent ‘realizations’ arising from different definitions presented in text books.
2015
Pittsburgh, Pennsylvania
Over the last decade, undergraduate mathematics researchers and curriculum developers have generated inquiry-oriented curriculum materials for courses from calculus through abstract algebra. These materials present a number of challenges for implementation, and as such, an instructional quality measure becomes a necessary requirement for making sense of student learning in these classrooms. The work here represents an initial attempt to define and map the domains of inquiry-oriented teaching. Specifically, classroom video data will be analyzed using a conjectured list of critical components of inquiry-oriented teaching. This analysis will be used to 1) refine/test the initial characterization of inquiry-oriented teaching, and 2) generate examples of how these critical components are actualized during instruction.
2015
Pittsburgh, Pennsylvania
Understanding students’ perspectives about their mathematics classroom experience is important for supporting students’ learning. As part of a larger study, 15 students enrolled in enrichment sections of a developmental mathematics course were interviewed to explore students’ experiences and dispositions. We highlight areas that students described as helpful or not and the ways in which students’ framed themselves as having a “productive disposition” (National Research Council (NRC), 2001). Preliminary findings suggest students found activities most helpful when they thought that the activity might help them be successful on other activities and that they described themselves as having many characteristics of a productive disposition. A detailed analysis of these descriptions, however, uncovered narrow interpretations of “understanding” and “making sense” in mathematics
2015
Pittsburgh, Pennsylvania
Prior research has shown how the multiplicatively-based summation conception (MBS) is important for making sense of definite integral expressions in science contexts. This study attempts to accomplish two goals. First, it describes introductory lessons on integration from two veteran calculus teachers as a way to possibly explain why so few students draw on the MBS conception when making sense of definite integrals. Second, it reports the results from a design experiment intended on promoting not only the construction of the MBS conception, but its priming for activation when students see and interpret definite integrals expressions.
2015
Pittsburgh, Pennsylvania
Prior research has documented several conceptualizations students have regarding the definite integral, though the conceptualizations are largely based off of single-variable integral expressions. No research to date has documented how students’ understanding of integration becomes generalized for multivariate contexts. This paper describes six conceptualizations of multivariate definite integrals and how they connect to students’ prior conceptions of single-variable definite integrals.
2015
Pittsburgh, Pennsylvania
The argument for the importance of proving and of proof in the teaching and learning of mathematics has been repeatedly made by mathematics education researchers and by policy documents. There is also considerable research examining the existence of a gap in the proving and proof-constructing abilities of “novices” and “experts” in mathematics. However, considerably less research examines the nature of what constitutes expertise in proving mathematical statements, specifically with regard to the use of the individuals’ mathematical knowledge. This study uses grounded theory methods to examine “expert” and “novice” mathematicians in the process of proving mathematical statements. The result reported here focuses on the differences in the use of and search for counterexamples by the two populations. More specifically expert provers seem to value the unsuccessful searches for counterexamples, as well as the finding of valid counterexamples.
2015
Pittsburgh, Pennsylvania
We report on a case study aimed at researching the social interactions of a classroom focusing on the certainty of mathematical claims and justifications. Blending Harel and Sowder’s (1998) concept of “proof schemes” with Yackel and Cobb’s (1996) “sociomathematical norms,” we aim to expand on Fukawa-Connelly’s (2012) research on sociomathematical norms of proof presentations. Preliminary analysis of classroom interaction and student interview transcripts from a proof-based, axiomatic geometry course suggests the presence of sociomathematical norms related to argumentation that lie outside of proof validation that facilitate renegotiating proof schemes.
2015
Pittsburgh, Pennsylvania
Effective educational video games have been both created and researched in the K-12 literature (e.g., Riconscente, 2013), yet instructional games created for undergraduate mathematics and the relevant research in their effectiveness is scarce. As a team of researchers and game developers, we created an instructional video game, named Mission Prime, based on the concept of optimization in calculus. The game brings players into a visually complex and engaging world in which they perform tasks involving essential calculus content (in this case, optimization). We aimed to research the effectiveness of the game as compared to both homework and to no treatment and believe that, if given an environment that has visual-spatial manipulatives and no punishment for incorrectness, students could enhance their understanding of the mathematical content.
2015
Pittsburgh, Pennsylvania
The actions of mathematics teacher educators (MTEs) are an underrepresented area of mathematics education research. Likewise, researchers have argued that a pressing area of need in mathematics education is investigating how to support students’ covariational reasoning. The purpose of this proposed study is to investigate an MTE’s use of technology to engender covariational reasoning in a content course designed for pre-service teachers. Through the lens of Carlson et al.’s (2002) covariation framework and related theories, I will analyze interview and observation data in order to provide insights into the ways in which an MTE conceptualizes the use of technology in his classroom as well as how he plans to implement it to engender covariational reasoning. Additionally, I will focus on the way in which the MTE implements the technology including how students in the classroom reason about quantities that vary through the instructor’s use of technology.
2015
Pittsburgh, Pennsylvania
For this study, we aim to answer the following questions: 1) What conceptual knowledge do students need to have to be able to correctly solve optimization problems? 2) What weaknesses do students demonstrate when solving optimization problems? 3) How can we address these weaknesses and improve the teaching of optimization problems in calculus? In this paper, we discuss preliminary findings from this study, focusing on the responses of four first semester calculus students as they solve a basic optimization problem during a semi-structured interview. In particular, we observe how students’ understanding of function and variable influences their understanding of optimization problems. We believe we may be able to use APOS theory (Asiala et al, 1996) as a lens for studying how students understand optimization problems and begin to explore that in this paper.
2015
Pittsburgh, Pennsylvania
Initial qualitative work with five community college calculus instructors indicated that these teachers not only modified their textbook when introducing the concept of the derivative, but also formally and informally evaluated their text. In addition, during classroom observations and interviews, these teachers did not distinguish clearly between the idea of the derivative as an object or an operator. This preliminary report details these findings and proposes questions for further research.
2015
Pittsburgh, Pennsylvania
Kilpatrick (1987) noted, “successful teaching, like successful communication, depends on having a good model of the other” (p. 17). Successful teaching of collegiate mathematics for PSMTs will necessitate understanding of what knowledge PSMTs may bring with them to the learning of collegiate mathematics and through what processes PSMTs construct their knowledge. Thus, in-depth analysis of PSMTs’ mathematics is of significant importance to our field for providing PSMTs with meaningful learning experiences in mathematics content courses. In this study, I intend to investigate how PSMTs construct their own knowledge of abstract mathematical notions (e.g., polynomial rings, irreducible polynomial, factorization, minimal polynomials) and, second, the difficulties that they encounter while constructing their knowledge of abstract mathematical notions.
2015
Pittsburgh, Pennsylvania
There is a dearth of research on students’ use of mathematical language, particularly when writing proofs at the undergraduate level. In this exploratory study, we analyze written student exams (N=149) from an introductory proof course in order to identify different ways in which students’ use of written mathematical language differs from mathematicians’ writing in formal mathematical settings.
2015
Pittsburgh, Pennsylvania
In an effort to understand ways to help students solve counting problems successfully, we conducted a paired teaching experiment in which two students reinvented four counting formulas by generalizing their work from an initial set of basic problems. Subsequent to reinventing these four formulas, they solved all but one counting problem correctly, regularly drawing upon outcomes and displaying a set-oriented perspective. In this paper, we report on the problem that they missed, which involved combinations (the Bits problem: How many 256-bit binary strings contain exactly 75 0’s?). We describe a key aspect of their activity that we refer to as combinatorial encoding of outcomes, and we use this language to analyze the student work. We discuss the importance of encoding as an informal way to articulate bijections, and we suggest avenues for future work and pedagogical implications.
2015
Pittsburgh, Pennsylvania
Learning a mathematical practice, such as problem solving, is different from learning mathematical content. Realizing that mathematical practices are a fundamental aspect of engaging in mathematical activity, we seek to better understand the nature of mathematical practices, as well as how they are perceived by those who teach them. In this paper, we explore these issues with university mathematicians. In particular, we focus on explaining how mathematicians think about, learn, and teach mathematical practices. We consider mathematicians’ interpretations of various mathematical practices and consider how those interpretations may influence their goals for instruction perspectives on student thinking. Specifically we seek to know how mathematicians understand, think about, and practically address the teaching and learning of mathematical practice.
2015
Pittsburgh, Pennsylvania
We develop a theoretical model for conceptualizing the restructuring of computational / numerical tasks, usually considered advanced, with a network of spatial visual representations designed to support geometric reasoning and conceptual development. Through our restructuring of the well-known “popcorn box problem,” we illustrate key developmental understandings related to optimization and rate of change, as well as the possible conceptual blends afforded by a networked spatial visual approach.
2015
Pittsburgh, Pennsylvania
Classroom teaching in multiple sections of Calculus I at a large comprehensive research university was observed and coded using the Teaching Dimensions Observation Protocol (TDOP). Within lecture-based methods, multiple teaching styles were identified ranging from low to moderate to high engagement, sometimes including desk work or group work. A sample population of students from all three engagement groups was followed for one year in order to analyze persistence rates into Calculus II and retention rates in a STEM major. No significant differences were found in the retention rates either at the University or in STEM majors across groups, with an average of 43% of STEM majors having switched out of STEM or having dropped out of the University after one year. However, the group experiencing higher engagement instruction in Calculus I was found to have significantly higher grades in Calculus I and also in Calculus II.
2015
Pittsburgh, Pennsylvania
We report on a research-based effort to make calculus conceptually accessible to more students while simultaneously increasing the coherence, rigor, and applicability of the content learned in the courses. Recent studies have indicated that an approximation and error analysis approach to curriculum and instructional design can support a productive and coherent conceptual foundation for students’ reasoning about concepts defined in terms of limit. In this study we explore the affects of such an approach to curriculum design systematically implemented in the form of 30 labs spread throughout the first two semesters of calculus. Data taken from pre-tests at the beginning of Calculus 1 and posttests near the end of Calculus 2 indicate conceptual gains above the gains previously observed from students taught without approximation curriculum.
2015
Pittsburgh, Pennsylvania
2015
Pittsburgh, Pennsylvania
We report on a study in which we observed 73 mathematics majors completing seven proving tasks in calculus. We use these data to empirically address several hypotheses from the undergraduate proving literature. The key findings from this study include: (a) Nearly all participants introduced diagrams and examples on multiple tasks, (b) few students relied predominantly on either semantic reasoning or syntactic reasoning, and (c) there was little correlation between one’s propensity to use examples or diagrams and one’s mathematical achievement, either on the proof-writing tasks or on GPA in advanced mathematics courses.
2015
Pittsburgh, Pennsylvania
With the rise of concept inventory style student assessment, focus has been placed on research-based refinement of assessment questions. Of equal importance is how one arrives at the tasks meant to reflect a given construct. This report will consider how this might be done in a research-based manner. The creation of a Group Concept Inventory (with particular attention to isomorphism) will be utilized to illustrate the approach. The methodology attempts to add rigor to vague suggestions of using textbooks, consulting experts, and referencing literature when developing assessment tasks. Various aspects of understanding concepts and attention to which concepts are essential to group (at an introductory level) will be discussed. The methodology and accompanying example will detail a Delphi process for expert consensus, a narrative and exercise textbook analysis and a thorough literature exploration.
2015
Pittsburgh, Pennsylvania
Lampert et al. (2010) define ambitious teaching as teaching designed to achieve the ambitious goals of conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and productive dispositions. They argue that this kind of teaching necessarily involves actively engaging students. Yet, lecture continues to dominate introductory college calculus courses throughout the country. In this poster, we draw on data from an national project (Characteristics of Successful Calculus Programs) to explore both national trends regarding ambitious teaching practices in calculus and present two case studies that give some important insights into how department-wide ambitious teaching can be instituted and sustained.
2015
Pittsburgh, Pennsylvania
Currently, political and economic demand for students graduating with Science Technology Engineering and Mathematics (STEM) degrees is high, but unfortunately, a large percentage of students switch to non-STEM majors in the first year of study. Roadblock courses, such as Calculus I, can contribute to poor retention rates due to classroom environment and instructor practices. Current research suggests recitation sessions (or labs) led by teaching assistants (TAs) can positively impact student retention rates. This study investigates the role of labs in Calculus I instruction. Through classroom observation the researcher investigated the practices of TAs and through interviews the researcher explored how beliefs about the purpose of Calculus I labs by the lead instructor, TA, and students compared to one another as well as to the practices observed. Preliminary findings on the alignment of participant views and classroom practices will be presented, and implications for increasing student retention rates will be discussed.
2015
Pittsburgh, Pennsylvania
Prior research in linear algebra education has focused on documenting and understanding the difficulties students have with specific topics. In more recent years, the research has started to shift towards developing instructional methods to address these issues. In this study, I explore the ways in which two instructors implement inquiry-oriented materials focused on span and linear (in)dependence. One of the instructors had prior experience with these materials and the other did not. Through an analysis of video recordings of these classes, I use the Inquiry-Oriented Discourse Moves framework to analyze how each instructor conducts whole-class discussion and the affordances these discussions provide their students.
2015
Pittsburgh, Pennsylvania
Model analysis is a quantitative research method used in physics education research to analyze and interpret the meaning of students’ incorrect responses on a well- designed research-based multiple-choice test. We have adapted this method to study students’ understanding of function composition when functions are represented graphical. Model analysis accounts for the fact that students may hold more than one idea or conception at a time, and may use different ideas and concepts in response to different situations. It is uniquely suited to study students’ understanding of function composition, as students often hold multiple, sometimes conflicting misconceptions on function composition, which they may use at different times. Model analysis can capture information on self-consistency of a student’s responses. We collected data from a calculus class before and after the class reviewed composition of functions. We find that model analysis offers insights not offered by traditional statistics.
2015
Pittsburgh, Pennsylvania
This study investigates the effects of using spreadsheets in Business Calculus. Both the computer and non-computer sections were taught using reformed curricula that focused on business applications, the use of realistic data sets, and conceptual understanding. This study compares student attitudes towards mathematics in both spreadsheet and non-spreadsheet sections as measured by pre- and post- surveys, student interviews, and examination data.
2015
Pittsburgh, Pennsylvania
This study reports on how ten undergraduate students enrolled in business calculus reasoned about marginal change (marginal cost, marginal revenue, and marginal profit) while engaged in a task-based interview followed by a semi-structured interview. The study had two major findings: (1) students had difficulty distinguishing between marginal cost and approximate marginal cost and (2) students conflated marginal cost and marginal revenue with total cost and total revenue respectively. For future research, we might consider investigating how students’ understanding of marginal change impacts their ability to solve optimization problems situated in the context of cost, revenue, and profit.
2015
Pittsburgh, Pennsylvania
Students’ thinking about quantities that vary in tandem remains an important area of mathematics education research due to its implications for student success in mathematics. In this paper, we expand on a way of thinking about covarying quantities, called bidirectional reasoning, in ways not detailed in prior research. A student thinking bidirectionally understands two quantities varying so that the conceived relationship does not have an inherent dependency; the student understands and anticipates that quantities exist and covary simultaneously. After describing bidirectional reasoning and connecting to informing theories, we draw from our work with undergraduate students to illustrate a student reasoning bidirectionally. Because this paper serves as a (re)introduction to bidirectional reasoning and relationships, we close with potential implications of students’ bidirectional reasoning, and we hypothesize productive lines of future research.
2015
Pittsburgh, Pennsylvania
We describe a construct called shape thinking that characterizes individuals’ ways of thinking about graphs. We introduce shape thinking in two forms–static and emergent–that have materialized in our work with students and teachers over the past two decades. Static shape thinking entails thinking of a graph as an object in and of itself, and as having properties that the student associates with learned facts. Emergent shape thinking entails envisioning a graph in terms of what is made (a trace) and how it is made (covarying quantities). We provide illustrations of the shape thinking forms using examples from data that we have gathered with secondary students, teachers, and undergraduate students. We close with future research and teaching directions with respect to students’ shape thinking.
2015
Pittsburgh, Pennsylvania
This poster reports on a study of practices that mathematics professors use to grade proofs. In an initial study, four mathematicians evaluated and scored six proofs of elementary theorems written by students in a discrete mathematics or geometry course. The results indicated that, while the professors generally agreed in their overall evaluations of the proofs, the scores varied substantially. A follow-up study delved more deeply into the reasons for the spread in the scores. This poster presents four reasons why the professors did not always agree in their scoring of the proofs: (a) performance errors, (b) disposition toward grading, (c) judgments about the student’s level of understanding and the seriousness of errors, and (d) contextual factors.
2015
Pittsburgh, Pennsylvania
In this project, we form, describe, and study groups (or clusters) of students based on their academic history prior to their first semester in college. These clusters allow us to examine the effect of gender on a student’s academic career decisions, such as course and major selection, while controlling for the level of preparation. We begin with an overview of standard hierarchical clustering and discuss the pitfalls of a straightforward application with our data. We then describe how to adjust the technique in order to form stable clusters. Using these clusters, we find that course selection and STEM retention are related to a student’s gender, with female students more likely to leave STEM early than male students with the same level of preparation and college grades.
2015
Pittsburgh, Pennsylvania
Structure sense is foundational to mathematical thinking. This report explores high school math teachers’ meanings for the substitution principle, a sub-category of structure sense that research previously identified as sources of difficulty for students. A focus on meanings reflects our belief that teachers’ meanings directly impact the mathematical meanings students develop. We suggest ways of thinking that could lead to various response types as a resource for teacher educators to design professional development targeting improved structure sense for teachers.
2015
Pittsburgh, Pennsylvania
Students learn the mathematical operation of subtraction beginning in elementary school, along with key vocabulary to talk about that operation. However, the meanings that students develop for the word “difference” continue to play a role well into students’ study of undergraduate mathematics. In particular, a meaning for “difference” as representing a change in a quantity is essential to understanding and communicating about foundational ideas in Calculus. In this preliminary report, we consider meanings about the word “difference” held by calculus students as revealed on a pre-test in an on-going study designed to explore Calculus students’ structure sense. We further propose potential consequences for those meanings and describe methods to be used in data collection for the remainder of the Fall 2014 semester.
2015
Pittsburgh, Pennsylvania
This qualitative study is an investigation of the teaching philosophies of beginning mathematics graduate teaching assistants (MGTAs). The study considered the cases of two domestic and two international MGTAs. Three teaching philosophy statements from each of the participants were collected at three different stages of a semester-long teaching assistant preparation program (pre-service phase). Three one-on-one interviews were conducted with each participant in the following four semesters (in-service phase) after the conclusion of the pre-service preparation program course. These audio-recorded interviews were transcribed and analyzed using the constant comparative method. Beginning teaching philosophies of these participants and how their philosophies changed over time, during both the pre-service and the in-service phases, will be briefly discussed. The factors that influenced their teaching philosophies in both phases will also be discussed.
2015
Pittsburgh, Pennsylvania
We report the results of a two-year study of the effect of clickers in the large-lecture format through a variety of metrics. These metrics include specific quiz scores (including both conceptual and applicational questions), as well as pass rates (both for the general student body as well as for males and females specifically). We include statistical basis for our findings.
2015
Pittsburgh, Pennsylvania
This study investigates the nature of preservice elementary teachers’ knowledge of content and teaching (KCT) through responses to hypothetical student scenarios. Participants demonstrated two types of KCT: (1) specific KCT, which drew from the specific mathematics of the student scenario and most resembles the construct described by Ball, Thames, & Phelps (2008); and (2) general KCT, which presented itself as “canned” mathematical pedagogy. Examples and influences upon each type of KCT are explored to promote discussion on the differentiation of KCT and for further refinement of the concept of general KCT.
2015
Pittsburgh, Pennsylvania
Research from an action-object perspective is well positioned to benefit from the emerging field of mathematics educational neuroscience. In this theoretical paper, we review some relevant findings from neuroscience studies and interpret them from an action-object perspective. This interpretation demonstrates a strong alignment of action-object theories and neuroscience findings, thus affirming many aspects of action-object perspectives on mathematical development. Neuroscience also serves to further elaborate and generalize reflective abstraction—the basis for action-object theories—as a mechanism for mathematical development. Specifically, we can begin to understand the changes in neural functioning associated with the objectification of action. This understanding helps explain some of the limitations teachers experience when attempting to provoke and support students’ constructions of actions and objects.
2015
Pittsburgh, Pennsylvania
During our research into an inquiry-based-transition to proofs course we observed that several students used truth tables in unique ways. Typically the students translated mathematical statements into propositional logic formalism and used the truth tables as tools in their mathematical activity. Student activity varied from using truth tables to show that a statement was a tautology, to obtain conviction in the truth of the statement, to demonstrate equivalence, or to formulate a conjecture. We hypothesize that the unique use of truth tables emerged because it was the classroom community’s responsibility to socially negotiate what counted as a proof. We intend to present some of our preliminary findings, and inquire as to what further research into these cases might be worthy of pursuit.
2015
Pittsburgh, Pennsylvania
Researchers continue to emphasize the importance of examining students’ reasoning when constructing situations involving numerous quantities and relationships (e.g., quantitatively rich situations). To explore student reasoning in such situations, I conducted a semester-long teaching experiment with two mathematics education undergraduate students. The teaching experiment sessions were focused on providing the students repeated opportunities to conceptualize quantitatively rich situations. In this proposal, I explore a few themes that emerged through analyses of their activity, characterizing their thinking during their construction of such situations. For instance, the order in which a student coordinated two quantities (e.g., coordinating a change in quantity A then a corresponding change in quantity B versus coordinating a change in quantity B then a corresponding change in quantity A) emerged as critical to their images and their representations of the situation. This and other findings provide important insights into ways students’ reason quantitatively and covariationally.
2015
Pittsburgh, Pennsylvania
Researchers have indicated that pre-service teachers (along with in-service teachers and college students) typically do not develop productive meanings for function and function inverse. In order to explore pre-service teachers’ inverse function meanings further, we conducted clinical interviews with 25 pre-service teachers. In this paper, we include a summary of previous research concerning individuals’ inverse function meanings as well as a description of the methodology and theoretical framework we used when making sense of the pre-service teachers’ activities. We present and interpret data highlighting the techniques the pre-service teachers used during the tasks. We present our interpretation of these results in relation to common curricular approaches to inverse function. We conclude with implications from our findings and areas for future research.
2015
Pittsburgh, Pennsylvania
The introductory linear algebra course provides unique challenges to many undergraduate students. The abstract nature of the course and immense amount of vocabulary offers a stark contrast to the procedural mathematics encountered in calculus and high school mathematics courses (Carlson, 1993; Dorier & Sierpinska, 2001). Many concepts in linear algebra are inherently connected. These connections can be made explicit by the use of a unifying and generalizing concept, defined by Dorier (1995). Examples of a unifying concept include the Invertible Matrix Theorem as presented by Lay (1994). In my study, I considered two potential unifying concepts, pivots and solution sets of matrix equations and asked how these concepts influenced student understanding of linear algebra.
2015
Pittsburgh, Pennsylvania
Silence has been important to many disparate traditions, notably, Zen Buddhism, and Taoism. But it has received relatively little treatment in the mathematics education literature. This paper attempts to begin a conversation on silence, its good and bad uses, and raises the question of whether silence may be an important aspect of mathematics activity. In order begin an answer to that question, it analyzes the contribution that a particular group member's silence produced for the group, leading both to a correct solution, and to a valuation of her group-mate's arguments.
2015
Pittsburgh, Pennsylvania
Function has been shown to be an important, but difficult concept for students to master (Carlson, Oehrtman & Engelke, 2010; Dubinsky & Harel, 1992). Through a clinical interview with a preservice mathematics teacher, I characterize the ways in which her function definition is able to account for novel relationships between quantities. Utilizing APOS theory, I find that though she is able to exhibit a process view of function, the student struggles to reconcile her definition of function with her intuitions about domain/co-domain and causation. The research is part of a larger study examining the ways in which preservice teachers define function affect their ability to accommodate novel contexts and representations.
2015
Pittsburgh, Pennsylvania
This study examined two lessons taught in parallel in a real analysis course by two different teaching assistants. The lessons were based on a single lesson-plan but decisions the instructors made prior to and during class took the lessons in substantially different directions. In this paper we focus on one instructor and describe some of the ingenious ways by which he adapted the written curriculum and introduced mathematical practices and ways of thinking in his lesson. We apply Schoenfeld's resources, orientations and goals framework to highlight relationships between the instructors' beliefs and practices, and to propose explanations as to why this instructor implemented the written curriculum and addressed the mathematics the way he did. On the basis of this analysis we suggest a new perspective on the nature of the process by which university instructors make sense of the curriculum, derive and prioritize goals and form their lesson-images.
2015
Pittsburgh, Pennsylvania
Recent discussions in the field have explored proofs’ explanatory power. Such research, however, focuses on how a written proof might convey explanation. I present a conjecture that individual proof activity (the development of proofs) might, itself, have explanatory power. I then discuss one student’s (John’s) activity related to proving that the centralizer for a fixed element in a group (the set of elements that commute with the given element) is a subgroup and how this activity informed his understanding of inverse. During an individual interview, John developed a lemma claiming that the left- and right- inverses of an element are the same element, his proof of which contradicted his previous ways of thinking about inverse. I analyzed John’s proof activity using Aberdein’s (2006) extension of Toulmin’s (1979) model for argumentation in order to better organize his activity, providing an example of how proof activity might itself be explanatory.
2015
Pittsburgh, Pennsylvania
A challenge in mathematics education research is to coordinate different analyses to develop a more comprehensive account of teaching and learning. We contribute to these efforts by expanding the constructs in Cobb and Yackel’s (1996) interpretive framework that allow for coordinating social and individual perspectives. This expansion involves four different constructs: disciplinary practices, classroom mathematical practices, individual participation in mathematical activity, and mathematical conceptions that individuals bring to bear in their mathematical work. We illustrate these four constructs for making sense of students’ mathematical progress using data from an undergraduate mathematics course in linear algebra.
2015
Pittsburgh, Pennsylvania
A full examination of learning or developing systems requires data analysis approaches beyond the commonplace pre-/post-testing. Drawing on graph theory, three particular approaches to the analysis of data – based on adjacency matrices, affiliation networks, and edit distances – can provide additional insight into data. Data analysis methods based on adjacency matrices demonstrate that learning is not unidimensional, and that learning progressions do not always progress monotonically toward desired understandings and also provide insight into the connection between instruction and student learning. The use of affiliation networks provides insight into how students’ prior knowledge relates to topics being studied. Careful use of edit distances indicates a likely overestimate of effect sizes in many studies, and also provides evidence that concepts are often created in an ad hoc manner. All of these have implications for curriculum and instruction, and indicate some directions for further inquiry.
2015
Pittsburgh, Pennsylvania
Calculus is a foundational sequence in mathematics and many client disciplines, such as physics and engineering. For student success both in mathematics and in these client disciplines, the mathematical background provided must go beyond simple instrumental or procedural skills to a deeper level. For success in higher-level mathematics, students must delve to the level of logical understanding, being able to articulate logical connections between two mathematical concepts. In this study we analyze students’ abilities to explain the connection between the limit definitions of derivative and definite integral, and their common geometric interpretations involving slope and area. We also determine whether a group exercise early in the term which reinforces the connection between the derivative and slope enhances students’ written responses concerning the connection on the final exam.
2015
Pittsburgh, Pennsylvania
The purpose of this study is to explore students’ activities while they construct existence proofs. We focus on three undergraduate students who completed a transition-to-proof course, and analyze their constructions of existence proofs. The results indicate that students’ activities for existence proofs were associated with their interpretations of a given statement, strategic knowledge for their proofs, and proving frames recruited. In addition, we discuss how the students’ conceptions of proof also play a role in their construction of existence proofs.
2015
Pittsburgh, Pennsylvania
This paper extends the theoretical framework for exploring student understanding of the concept of the derivative, which was developed by Zandieh (2000). We expand upon the concept of a physical representation for the derivative by extending Zandieh's map of the territory to provide higher resolution in regions that are of interest to those operating in a physical context. We also introduce the idea of "thick" derivatives, which are ratios of small but not infinitesimal changes, which are practically equivalent to the true derivative.
2015
Pittsburgh, Pennsylvania
This study examines similarities and differences in the Advanced Placement and college calculus experience from the student perspective to characterize how taking AP Calculus in high school relates to student success in a college calculus course. Fourteen first–semester college students who had taken the AP Calculus exam were interviewed about their perceptions of, and experiences in the courses. The Academic Performance Determinants Model (Credé & Kuncel, 2008) was used to develop an interview protocol. Qualitative analysis of the interviews revealed four categories of themes about the students’ experience: 1) Students’ study approaches in the respective classes, 2) Students’ self–efficacy and metacognition, 3) The Class format’s effect on student success, and 4) Students’ beliefs about the cognitive demand of the course. All the themes, their implications for those teaching in AP and college calculus, and the need for further research are presented.
2015
Pittsburgh, Pennsylvania
This paper presents ongoing research whose goal is to study theoretical frameworks that support teaching practices carried out in courses that are part of a Bachelor's Degree in mathematics which use information and communication technology resources - such as blogs, social networks and virtual learning environments - for the teaching and learning of mathematics content that is part of the curriculum in colleges and universities. The outline presented here refers to the ongoing study in the Mathematics, Culture, Art and Technology line of research being done by the research group THEM (Spices of History in Mathematics Education) about connectivism as a theoretical approach to teaching and learning.
2015
Pittsburgh, Pennsylvania
Existing research shows there is a strong correlation between spatial reasoning and mathematics achievement in K-12. Is spatial reasoning important then for succeeding in undergraduate and graduate mathematics? I interviewed four (N=4) young mathematicians about their conceptualizations of spatial reasoning and where they potentially saw spatial reasoning in undergraduate and graduate mathematics. Two participants identified themselves as geometers/topologists and the other two identified as algebraists. The two geometers/topologists emphasized understanding 2D representations of 3D shapes and the two algebraists emphasized the notion of dynamics, i.e. mental movement. Among graduate fields of mathematics, geometry/topology was cited most frequently as being spatial and algebra was cited the least. In undergraduate mathematics, all four participants drew connections to multivariable calculus and discussed examples from their teaching of multivariable calculus. This work has some implications for teaching multivariable calculus and discerning the relationship between spatial reasoning and mathematics.
2015
Pittsburgh, Pennsylvania
There is a considerable amount of mathematics education literature on creativity (e.g., Torrance, 1966; Balka, 1974; Silver, 1997), yet there is little discussion of mathematical creativity in undergraduate mathematics education. Specifically, to our knowledge, there is no literature on mathematical creativity in the proving process. We attempt to contribute to the literature by developing a framework in which to discuss mathematical creativity in proving, Creativity-in- Progress Rubric (CPR) on proving. Through previous rubrics on creativity (Leikin, 2009; Rhodes, 2010) and interviews with both mathematicians and undergraduate students, we claim that there are three aspects of creativity in proving: Making Connections, Taking Risks, and Creating Ideas. We demonstrate how to use the rubric with an example of a student’s proving process. Finally, we give future research considerations of using the rubric in proof-based classrooms.
2015
Pittsburgh, Pennsylvania
This theoretical paper suggests a perspective for understanding undergraduate proof construction based on the ideas of conceptual and procedural knowledge, explicit and implicit learning, behavioral schemas, automaticity, working memory, consciousness, and System 1 and System 2 cognition. In particular, we will discuss proving actions, such as the construction of proof frameworks that could be automated, thereby reducing the burden on working memory and enabling university students to devote more resources to the truly hard parts of proofs.
2015
Pittsburgh, Pennsylvania
The purpose of this study is to examine how individual college students reason through tasks using trigonometric functions and translate among different types of representations of trigonometric functions across various mathematical tasks.
2015
Pittsburgh, Pennsylvania
This paper reports on an investigation of fifteen second-semester calculus students’ understanding of the concept of parametric function, as a special relation from a subset of R to a subset of R2. A substantial amount of research has revealed that the concept of function, in general, is very difficult for students to understand. Furthermore, several studies have investigated students’ understanding of various types of functions. However, very little is known about how students reason about parametric functions. Employing APOS theory as the guiding theoretical perspective, this paper describes how students reason about parametric functions given in the form p(t) = (f(t),g(t)). One common misconception that was observed among students is addressed.
2015
Pittsburgh, Pennsylvania
We present results from a study that examined the use of journal writing by undergraduates in an Honor’s Number Theory course and how the journals supported students’ learning, attitudinal changes, and proof writing. The 17 undergraduates in this course submitted weekly journals online to their instructor and reflected on their mathematical learning. The instructor provided comments to each of the students’ journal submissions that informed him of each student’s successes, challenges, issues, and questions. We analyzed the journals and share our preliminary findings on what the journal writing revealed about students’ learning and how their mathematical understanding developed over a semester. We include results of the pre-post survey of student attitudes toward mathematics along with interviews of 5 of the students that give additional insight into their experiences in the course. Future work will examine the uses of journals in other courses, and different ways journals support student learning
2015
Pittsburgh, Pennsylvania
The use of technology in teaching and learning has been explored by many mathematics education researchers. It is believed that, the technology itself is not a magic bullet to attract students to mathematics and more importantly help them to learn.
2015
Pittsburgh, Pennsylvania
In this paper we examine an Abstract Algebra professor and one of his students’ thought processes simultaneously as the class was moving toward the proof of the Fundamental Theorem of Galois Theory. We employed Tall’s theory of three worlds of mathematical thinking to trace which route (embodied, symbolic, formal) the mathematician was choosing to take his students to the formal world. We will discuss the pedagogical challenges of proving an elegant theory as the events unfolded.
2015
Pittsburgh, Pennsylvania
In this paper we present a mathematics professor’s thought processes while teaching Calculus I, as shared through her teaching diaries and later discussed in weekly meetings with a team of two other mathematicians, a mathematics educator, and a cognitive psychologist over the period of a semester. We examine the way she balanced formal and symbolic thinking while encouraging embodied thinking throughout her lectures. Moreover, we will discuss some data obtained from students through interviews, a questionnaire, and end-of-semester course evaluations.
2015
Pittsburgh, Pennsylvania
This presentation reports on research investigating what mathematical constructs and ways of understanding constitute a formal understanding of limit. This work builds primarily on the genetic decomposition of Swinyard and Larsen (2012), which itself was modified from Cottrill, et al. (1996). Ten undergraduate students were interviewed in a semi-structured, clinical setting. While analysis is in the preliminary stages, the data suggest that the types of tasks given have a significant influence on whether or not formal understanding is demonstrated.
2015
Pittsburgh, Pennsylvania
To produce more STEM graduates in the U.S., improving student success in calculus is crucial; previous research suggests that students with a proclivity to visualize when solving mathematics tasks are not the “star” students in mathematics classrooms. A study of undergraduate curriculum also found that common calculus tasks reinforce procedural understanding. Since incorporating dynamic visualization (DV) provides a possible tool for increasing understanding, we investigate the role of DV in calculus learning at the university level. We examine student understanding of derivative as a rate of change by comparing student experiences when exploring with DV software or engaging with static tasks in individual interviews collected in four episodes over one semester on four students identified as visualizers and five as non-visualizers. Comparisons reveal the emergence of cognitive conflict and its resolution for students encountering the DVs but this resolution is not evident for those only engaged in static work.
2015
Pittsburgh, Pennsylvania
This paper reports the results of a series of task-based clinical interviews I conducted to examine how a secondary mathematics teacher’s understandings of angle measure afford or constrain his capacity to bring his mathematical knowledge to bear in the context of teaching. The results suggest that the teacher, David, possessed two complementary but conceptually distinct ways of understanding angle measure that he was not consciously aware of having. As a result David was unable to strategically employ his two ways of understanding in novel problem-solving situations and was unable to leverage his understandings in the context of teaching. I also discuss the effectiveness of an instructional intervention designed to support David in becoming aware of his understandings and conclude that engaging teachers in experiences that promote reflected abstraction is one way of supporting them in transforming their mathematical knowledge into a pedagogically efficacious form.
2015
Pittsburgh, Pennsylvania
Mathematical creativity has been an object of discussion in mathematics for some time (Borwein, Liljedahl, & Zhai, 2014). Though there are recent efforts to include creativity in K-12 education agendas (Askew, 2013) and a number of research articles in mathematics education literature about creativity focus K-12 (e.g., Silver, 1997), there is little research in undergraduate mathematics education about creativity. Our study aims to address this issue by examining what mathematicians and undergraduate students think about creativity in proving. We coded six mathematician and eight student interviews using the Creativity in Progress Rubric (CPR) on proving created by the research group. A majority of the students’ responses were of the Creating Ideas category of the rubric, while mathematicians were more balanced. We claim that the other two categories (Taking Risks and Making Connections) might need further explicit discussion in classrooms if the transition from student to mathematician is desired.
2015
Pittsburgh, Pennsylvania
Historical number systems were leveraged to build preservice teachers (PTs) understanding of base 10. Variation theory states that you cannot know something if all you know is that one thing. Thus to understand numbers and operations in base 10 PTs need to experience number and operations within and beyond base 10. In this study we examined how historical numbers systems can be used to allow PTs to build a conceptual understanding of numbers and operation in base ten by comparing and contrasting various number systems. Thirty-six PTs concepts of number were identified before and after they work through a series of tasks situated in various number systems and compared and contrasted across the systems. Almost all PTs developed a more sophisticated conception of number though these experiences.
2015
Pittsburgh, Pennsylvania
Concept inventories have become an increasingly popular way to measure conceptual understanding in STEM disciplines. The Force Concept Inventory (FCI; Hestenes, Wells and Swackhamer, 1992) was the first concept inventory to make a significant impact in the undergraduate education community, becoming widely used and significantly affecting the teaching of introductory hi physics. The FCI paved the way for the broad application of analyzing student conceptual understanding of the basic ideas in a STEM subject area (Hake, 1998a, 2007; Hestenes & Wells, 1992; Hestenes et al., 1992); concept inventories have been written for biology, chemistry, and astronomy.
2015
Pittsburgh, Pennsylvania
As part of work on student understanding of the Fundamental Theorem of Calculus (FTC) and definite integrals, we incorporated a technique known as eye tracking to investigate how students attribute their visual attention while answering graphically-based questions. The direction and duration of eye gaze of 17 students was recorded in real time. We analyzed the total proportion of time spent on various question domains (lexicons, equations & symbols, graphs, and question options) as well as on various relevant and irrelevant features of the graphs. We found that the students who responded correctly spent more time on relevant graphical features, whereas those responded incorrectly spent more time on irrelevant graphical features. We also found that student visual attribution depends on types of representations and notations provided in the questions. Most of the eye-tracking results corroborate previously reported written and interview results on student application of the FTC across the mathematics-physics interface.
2015
Pittsburgh, Pennsylvania
In this poster I would like to share my ideas on how to help students overcome math anxiety, exam anxiety in particular. Math anxiety is reported to be a major learning obstacle for many students at various levels of learning mathematics; students worry about outcomes of in-class assessments, especially exams. Instructors typically place a lot of weight on exams, so students fear that if they do poorly their grade will go down. This is known as exam anxiety; not necessarily math test anxiety, but putting the two together would certainly multiply students’ stress. In this poster I would like to illustrate how videos can be utilized to ease math test anxiety and help improve students’ overall performance in math classrooms.
2015
Pittsburgh, Pennsylvania
This study sought to describe the ideas professional mathematicians’ find useful in moving their arguments forward while constructing mathematical proof and the context surrounding the development of these ideas. Three research mathematicians completed real analysis tasks while thinking aloud in interview and independent settings recorded through Livescribe technology. Follow-up interviews were also conducted. Data were analyzed for perceived useful ideas and coded based on Dewey’s inquiry framework and Toulmin’s argumentation model. Toulmin argumentation diagrams were implemented to describe the evolution of the arguments, whereas Dewey’s inquiry framework helped to describe the context surrounding the development of the ideas. Preliminary findings show an active inquiry into forming a geometric understanding to develop warrants based on intuition and examples and then working to find backing that can be rendered into a formal argument.
2015
Pittsburgh, Pennsylvania
This research stems from efforts to infuse quantitative literacy (QL) in an online version of college algebra. College algebra fulfills Appalachian's QL requirement, and is a terminal course for most who take it. In light of the course’s traditional content and teaching methods, students often leave with little gained in QL. An online platform provides a unique means of engaging students in quantitative discussions and research, yet little research exists on online courses in the context of QL. The researcher’s course includes weekly news discussions as well as “messy” projects requiring data analysis. Students in online and face-to-face sections of the course took the QLRA (developed by the NNN) during the first and final weeks of the fall 2014 semester. There were significant statistical gains in the online students’ QLRA performance and mathematical affect but none for the face-to-face students. Implications of this include that project-based learning in an online environment is a promising strategy for fostering QL in terminal math courses.
2015
Pittsburgh, Pennsylvania
Common misconceptions that students have about tangent lines are well documented in the research literature. This study seeks to understand the efforts that instructors make to address these common misconceptions in their classroom instruction. Specifically, we looked at video data from classroom sessions of five instructors when they covered the graphical representation of derivative. Language and gestures instructors used as well as the graphical examples they provided to the students were analyzed.
2015
Pittsburgh, Pennsylvania
Student understanding of solution is central to success in much of mathematics, from basic algebra through linear algebra and differential equations. This research explores college algebra students’ understanding of solution. In particular, it explores student definitions of what it means for a number to be a solution to an equation and whether students can determine if a number is a solution to an equation. Results show that most of the students could determine if a given number is a solution to an equation but that fewer than half of the students could write a reasonable definition of solution. Categories of student responses are identified along with possible reasons for the misconceptions. These results have implications for teaching all levels of mathematics.
2015
Pittsburgh, Pennsylvania
In this study we use the lens of knowledge at the mathematical horizon to shed light on the underlying structural components of school mathematics content. We attend specifically to algebraic structures, identifying ways in which awareness of such structure may be transformative to mathematical knowledge for teaching. In particular, we analyze curricular content from elementary, middle, and secondary school mathematics with respect to the inherent structures that encompass and connect that material, as well as with specific attention to the opportunities afforded to teachers by their horizon.
2015
Pittsburgh, Pennsylvania
Fourteen secondary mathematics teachers were given a task-based interview in which they were presented with four mathematical tasks from secondary school mathematics that a real analysis course might prepare them to handle. We found that most participants could not answer these questions correctly. Some participants believed the answers to some of these questions could inform their teaching, but felt these topics were not included in their real analysis course. We suggest that explicitly discussing the corollaries relevant to secondary school mathematics in real analysis courses for prospective teachers might be a useful first step in helping future teachers see real analysis as relevant to their instruction.
2015
Pittsburgh, Pennsylvania
The derivative is an important foundational concept in calculus that has applications in many fields of study. Existing frameworks for student understanding of the derivative are largely descriptive in nature, and there is little by way of theoretical frameworks that can explain or predict student difficulties in working with the derivative concept. In this paper we combine Zandieh’s framework for understanding the derivative with “analogical reasoning” from psychology into the “merged derivative-analog framework.” This framework allows us to take the useful descriptive capabilities of Zandieh’s framework and add a layer of explanatory power for student difficulties in applying the derivative to novel situations.
2015
Pittsburgh, Pennsylvania
Linear algebra is widely viewed as pivotal yet difficult for university students, and hence innovative instructional materials are essential. The goals of this NSF-funded research project include producing: (a) student materials composed of challenging and coherent task sequences that facilitate an inquiry-oriented approach to the teaching and learning of linear algebra; and (b) instructional support materials for implementing the student materials. This poster will highlight the third unit of the IOLA (Inquiry Oriented Linear Algebra) materials that focus on a research-based approach to introducing eigentheory, change of basis, and diagonalization.
2015
Pittsburgh, Pennsylvania
Reasoning used by students as they verify trigonometric identities has not been investigated. Through analyzing students’ spoken explanations of their thought processes in clinical interviews, this current study explores why students choose certain expressions to begin manipulating and why certain manipulations or substitutions are performed as they verify the purported identity. Preliminary findings suggest that students may use beliefs about mathematics to inform and monitor their decisions, namely, the belief that mathematical answers are results of a simplification process. Thus, students spoke of the flow of verification problems in terms of a simplifying process. This belief may be imported from a cultural belief that being simpler is better and more desired; thus, mathematical tasks could validate the imported belief and in turn strengthen the belief about mathematics as simplification. To support this possibility, this paper will share student comments. Implications for instructional practice will also be suggested.
2015
Pittsburgh, Pennsylvania
In mathematics education research paradoxes of infinity have been used in the investigation of students’ conceptions of infinity. We analyze one such paradox – the Painter’s Paradox – and examine the struggles of a group of Calculus students in an attempt to resolve it. This study shows that contextual considerations hinder students’ ability to resolve the paradox mathematically. We suggest that the conventional approach to introducing area and volume concepts in Calculus presents a didactical obstacle. A possible alternative is considered.
2015
Pittsburgh, Pennsylvania
Studying families of functions is important for developing an understanding of the concept of function. This study utilized a computer algebra system (CAS) learning environment as it explored how students conceptualize rational functions. Using APOS theory as a theoretical framework, student artifacts from video, audio, and screen recordings were coded as either: action, process, or object conceptions of rational functions. The results of this study describe how students portray different conceptions of rational functions. This list is not exhaustive, but provides a starting point for identifying how students perceive of rational functions in a technological environment. Developing an understanding of how students conceptualize rational functions could lead to useful implications for teaching.
2015
Pittsburgh, Pennsylvania
One of the most important concepts in calculus is the derivative. Unfortunately, many studies have shown that students have trouble understanding derivatives, possibly due to deficiencies in knowledge about concepts learned prior to calculus. However, few studies have investigated how learning about other calculus topics affects students’ understandings of the derivative. In this study, qualitative methods were used to investigate the influence, or backward transfer effect, that learning about integration has on students’ prior understandings of derivatives. Semi-structured task-based interviews were conducted with four high school students before and after students received instruction on integration. Interview tasks involved finding derivatives and antiderivatives algebraically and graphically. Results from this study may show that learning about integration is another potential reason for why students have trouble understanding derivatives. Alternately, results may show that it is possible for teachers to help reinforce students’ understandings of derivatives through instruction on other calculus topics.
2015
Pittsburgh, Pennsylvania
One of the most difficult things for a monolingual teacher to decide is if errors by a student who is acquiring English language reflect a lack of mathematical understanding or some problems with English language. This paper buttresses this point with some anecdotal evidences from mathematical works of Preparatory Year University students in which students might be right, but the chances are high that the teacher, especially monolingual will wrong him. The result shows the dominance of students’ first language (Arabic) over the language of teaching and learning (English) while students are doing mathematics. This show how the language conflict can affect students’ assessment in mathematics.
2015
Pittsburgh, Pennsylvania
Lack of Proficiency in English is one of the major obstacles in students learning of mathematics in English medium universities in Saudi Arabia. Despite, this, some students are found doing exceptionally good in mathematics examinations conducted in English – a language that the students are not proficient in. In this paper, we share some of the challenges of learning mathematics in English language by this class of students, and the learning resources that help them to overcome the problems.
2015
Pittsburgh, Pennsylvania
The origins of linear algebra lie in efforts to solve systems of linear equations and understand the nature of their solution sets. In our experience, instructors of linear algebra see the work of teaching students to solve linear systems as the more straightforward and procedural portion of the course. We speculate that solving linear systems and interpreting their solution sets in fact entails hidden and significant challenges for students that are important for their later success in linear algebra, as well as their work in related STEM courses. In this paper, we examine final exam data from 69 students in an introductory undergraduate linear algebra course at a large public university in the southwestern US. Our analysis suggests that students are largely successful in representing systems of linear equations using augmented matrices, but that interpreting the row reduced echelon form of these matrices is a common source of difficulty.
2015
Pittsburgh, Pennsylvania
The purpose of this preliminary report is to introduce a new type of assessment instrument to the mathematics education research community and to reflect with our colleagues about the possible affordances and constraints of this instrument. The questions that comprise the instrument consist of a multiple choice (MC) stem followed by a series of options from which students choose explanations (E) that support their multiple choice response. We call this style of question multiple-choice with explanation (MCE). Our decision to use MCE style questions is informed by cutting edge work in physics education research (Wilcox & Pollock, 2013) and introduces an innovative idea for assessing student thinking in mathematics. Our work is part of a larger project in linear algebra; as such, the mathematical context of the assessment instrument is linear algebra. The format of the questions, however, could be used for other subject matter as well.
2015
Pittsburgh, Pennsylvania
We examined the proof-writing behaviors of six highly successful mathematics majors on proving tasks in calculus. We found that these students approached the proof writing tasks in two different ways. Three students, who we labeled as drillers, would develop a strong understanding of the statement they were proving, choose a plan based on this understanding, develop a graphical argument for why the statement is true, and formalize this graphical argument into a proof. The other three students, who we labeled as probers, would begin trying different proving approaches immediately after reading the statement and would abandon an approach at the first sign of difficulty. Despite being inconsistent with theories of effective problem solving in the mathematics literature, the probers were highly successful in their advanced mathematics courses and on the proving tasks in this study.
2015
Pittsburgh, Pennsylvania
This paper describes the evolution of a teaching technique for calculating volume using multiple integrals - from a physical to virtual model - in a Differential and Integral Calculus (CDI) course involving students in math and engineering programs at a public university in Brazil. This transition was made possible by the use of e-learning tools on the Moodle platform, which is used to support classroom activities. As a result of this technique, we saw the deepening of the concepts of parameterization taught in analytic geometry that were necessary to build the model using graphics software (virtual) as well as a change in the dynamic of sharing the technique among teachers and students through discussion forums in a continuous process.