PROCEEDINGS OF THE 20TH ANNUAL CONFERENCE ON RESEARCH IN UNDERGRADUATE MATHEMATICS EDUCATION
2017
San Diego, California
Number of pages: 1702
Number of papers: 266
- Aaron Weinberg
- Chris Rasmussen
- Jeffrey Rabin
- Megan Wawro
- Stacy Brown
2017
San Diego, California
Number of pages: 1702
Number of papers: 266
2017
San Diego, California
Real analysis is frequently a required course for prospective secondary mathematics teachers. However, most teachers view real analysis as unnecessary and unrelated to the work of teaching secondary mathematics. The purposes of this paper are to (i) explore why real analysis, as it is conventionally taught, is not helpful to many teachers, (ii) present a new instructional model for how the course can be taught to increase its relevance, and (iii) present a case study in which our instructional model was implemented in a real analysis course and led to productive changes in teachers’ actual pedagogical practice.
2017
San Diego, California
In this paper, we present a comparative case study of two students with different epistemological frames watching the same real analysis lectures. We show that students with different epistemological frames can interpret the same lecture in different ways. These results illustrate how a student’s interpretation of a lecture is not inherently tied to the lecture, but rather depend on the student and her perspective on mathematics. Thus, improving student learning may depend on more than improving the quality of the lectures, but also changing student’s beliefs and orientations about mathematics and mathematics learning.
2017
San Diego, California
We present a case study of Hugo’s construction of Euler diagrams to develop set-based meanings for mathematical conditionals. This episode arose in a teaching experiment guiding students to reinvent mathematical logic from their reasoning about meaningful mathematical statements. We intended for Hugo to develop a subset meaning for conditional truth. Hugo successfully identified and used this condition, but he also introduced another formally equivalent meaning for conditional truth. We discuss the shifts in his thinking necessary for developing set-based reasoning and how this case influenced our goals for logic learning.
2017
San Diego, California
In 2010, Charalambous published an article that examined the relationship between mathematical knowledge for teaching and task unfolding at the elementary level. As a result of this study, Charalambous evidence to support the claim that there is a positive relationship between a teacher’s MKT and the cognitive level of enacted task. Drawing upon this finding, the purpose of this study is to propose a new methodological approach examining mathematical knowledge for teaching at the undergraduate level. While this approach draws upon results concerning MKT at the K-12 level, it primarily focuses on examining undergraduate instruction through the lens of task unfolding and cognitive demand. To illustrate how this methodology can be used, the paper concludes by presenting two case studies that demonstrate how the methodology can be used to examine mathematical knowledge for teaching undergraduate Precalculus courses.
2017
San Diego, California
Proof by mathematical induction is arguably the most difficult proof technique for students to master. We explain this difficulty within an action-object framework. Specifically, we report on results from clinical interviews with two mathematics majors in which the first author administered tasks designed to elucidate each student’s understanding of logical implications as mental objects. We found that the framework explains much of the difficulty inherent in proof by induction, even the students’ struggles with hidden quantifiers.
2017
San Diego, California
This study examines students’ reasoning about eigenvalues and eigenvectors as evidenced by their written responses to two open-ended response questions. This analysis draws on data taken from 126 students whose instructors received a set of supports to implement a particular inquiry-oriented instructional approach and 129 comparable students whose instructors did not use this instructional approach. In this chapter, we offer examples of student responses that provide insight into students’ reasoning and summarize broad trends observed in our quantitative analysis. In general, students in both groups performed better on the procedurally oriented question than on the conceptually oriented question. The group of students whose instructors received support to implement the inquiry-oriented approach outperformed the other group of students on the conceptually oriented question and performed equally well on the procedurally oriented question.
2017
San Diego, California
Hypothesis testing is a key concept included in many introductory statistics courses. Yet, due to common misunderstandings of both scientists and students, the use of hypothesis testing to interpret experimental data has received criticism. With statistics education on the rise as well as an increasing number of students enrolling in introductory statistics courses each year, there is a need for research that investigates students’ understanding and curriculum effectiveness of hypothesis testing. This paper describes results obtained from a larger study designed to explore introductory statistics students’ understanding of one sample hypothesis testing. In particular, this paper explores students’ understanding of test statistic as a component of hypothesis testing. APOS Theory is used as a guiding theoretical framework. This paper focuses on three students’ understandings of test statistic when performing hypothesis tests on real world data.
2017
San Diego, California
The purpose of this study is to examine the characteristics of students’ visual reasoning in the context of evaluating statements about real-valued functions. We conducted clinical interviews with nine undergraduate students in which we asked them to evaluate several mathematical statements using graphs to explain their reasoning. In this paper, we focus on two Advanced Calculus students and the differences in their visual reasoning in these tasks. Our findings indicate that students’ visual reasoning accounts for key differences in their understandings of mathematical statements. In this paper, we introduce a visual reasoning framework which emerged from our data. We also provide examples from the two students to highlight the use of the framework to characterize students’ visual reasoning as value-thinking or location-thinking.
2017
San Diego, California
The research community shares a concern for students’ conceptual understanding of calculus and commonly advocates for student-centered approaches as a way to promote it. In this study, we investigated the effect of different instructional approaches on 151 undergraduate students’ conceptual understanding of differential calculus in context-specific, natural settings. We collected data on the pre- and posttest of the Calculus Concept Inventory in three classes. In one class, most of the time was dedicated to conceptually oriented problem solving, another class implemented practice problems for students, and the third class was a traditional lecture class. The results showed that there was no difference in students’ conceptual understanding of differential calculus controlling for their initial understanding. Thus, our findings do not support the research that advocates for student-centered instruction suggesting that the approaches’ implementation and contextual differences may be sources of variation in their effectiveness.
2017
San Diego, California
In the evaluation study presented in this paper, the authors compared the mathematical thinking of undergraduate students (as they responded to class work and interview prompts) who participated in an inquiry-based linear algebra course to a comparison group of students who participated in a traditional course.
2017
San Diego, California
The results of educational research studies are only as accurate as the data used to produce them. Drawing on experiences conducting large-scale efficacy studies of classroom-based algebra interventions for community college and middle school students, I am developing practice-based data cleaning procedures to support scholars in conducting rigorous research. The poster identifies common sources of data errors in mathematics education research and offers a framework and related data cleaning process designed to address these errors.
2017
San Diego, California
This paper presents the findings from a survey used to investigate how mathematicians perceive the genre of mathematical proof writing at the undergraduate level. Mathematicians were asked whether proof excerpts were unconventional in three contexts: undergraduate textbooks, what instructors write on the blackboard in undergraduate courses, and how students write in these courses. There are four main findings. First, participants found some potential breaches unconventional regardless of the context in which they occur. Second, mathematicians perceived the linguistic conventions in blackboard proofs and student-produced proofs differently in some cases. Third, textbook authors are expected to adhere to stricter norms than instructors and students when writing proofs. Fourth, there were potential breaches that the literature suggests were unconventional, which were not evaluated as unconventional by the mathematicians.
2017
San Diego, California
Despite concerted efforts on the part of educational policy makers, women are still underrepresented in the STEM fields. Researchers have shown that calculus plays a major role in this gender disparity since it requires spatial skills to success -- skills that women tend to utilize differently compared to men. However, previous studies have shown that spatial ability is malleable and spatial skills can be improved with training. This pilot study employed a form of spatial training in a third-term calculus course and measured the effects of this training on students’ calculus ability, spatial rotation ability, and cognitive style. Associations between cognitive style and task performance were also measured. Preliminary results indicate that spatial training did not significantly impact student performance on a calculus skills assessment or a test of mental rotations, but effects on students’ cognitive style were present.
2017
San Diego, California
Students’ conceptions about binary operations often reflect a lifetime of situated learning about concepts such as arithmetic operations and functions in their K-12 years. This prior knowledge base bleeds into their experience of binary operation in abstract algebra, leading students to have differing and incomplete notions of what constitutes a binary operation. We present a qualitative study in which we classify students’ conceptions of binary operation in terms of variation theory and concept image and definition. These frameworks helped us to focus on critical and noncritical aspects of binary operation found in student reasoning. Our results indicate that students’ enacted objects about binary operation fall into one of three domains (i.e. function meaning, arithmetic meaning, and structural meaning) and often depend on task context. Additionally, students’ reasoning reflected a set of critical aspects that diverge from the conventional critical aspects of binary operation.
2017
San Diego, California
Pass rates in US first-year undergraduate mathematics courses are abysmally low. However, recent studies have found that there are doctoral programs that have improved pass rates by focusing on improving instruction. In particular, these exemplary programs focus on interpreting student thinking and that this instructional shift creates a more positive experience for undergraduate students. To scale up this result, it is important to understand how instructors connect student thinking to teaching actions. The purpose of this case study is to examine how two graduate student teaching assistants developed the ability to connect student work to hypotheses about student thinking and then link these hypotheses to teaching actions. For this analysis, the researchers introduce a framework that has potential to help providers of professional development identify how to strengthen graduate students’ opportunities to learn from teaching, as well identify variations in graduate students’ use of undergraduate student thinking to inform teaching actions.
2017
San Diego, California
This study evaluated the outcomes of an intervention focused on developing mathematics graduate teaching assistants’ (GTAs’) skills of noticing and effectively responding to instances of student mathematical thinking that have significant potential to further students’ learning. Four GTAs participated in a semester-long intervention that included individual analysis and group discussion of video of undergraduate mathematics lessons. The MOST Analytic Framework (Stockero, Peterson, Leatham, & Van Zoest, 2014) was introduced to aid in these activities. The GTAs also completed a pre- and post-interview to document their real time noticing and an assessment of common content knowledge. Results indicate that the intervention was successful in improving the GTAs’ noticing skills in a variety of ways and in their ability to propose student-centered responses.
2017
San Diego, California
In the exploratory study presented in this paper, the authors aim to construct a model for the processes by which students in a multivariable calculus class conceptualize solid regions in three dimensions. We designed and recorded student work from two tasks in which students must decode a description of a solid figure and answer questions assessing the strength of their con- ception of the figure. Presented here are findings and common themes that emerged from the analysis of interviews and group work on two of these tasks, including five generalizable obser- vations about how students process three-dimensional information.
2017
San Diego, California
In an introductory linear algebra course, students are expected to learn a plethora of new concepts as well as how these concepts are connected to one another. Learning these connections can be quite challenging for students due to the vast number of connections and student inexperience with mathematical logic. The study reported here consisted of an investigation into how inquiry-oriented teaching methods could be employed in an attempt to create opportunities for students to develop mathematical connections in an introductory linear algebra course.
2017
San Diego, California
Much work has been done in recent years to study students’ formulations of formal limiting processes. One of the most common goals is to foster a productive understanding of the relationship between the error bound epsilon and the domain of the convergence; what is called a range-first perspective. My study examines an advanced calculus student’s understanding of the relationships involved in convergence of functions, and how his prior experience with limits influenced his understanding. I unpack his cognitive organization of the dependence relationships between epsilon, N , and x in functional convergence. This case study demonstrates the effects of a persistent understanding that epsilon depend on N in the convergence of sequences.
2017
San Diego, California
Part of a larger study of the development of teaching among novice college mathematics instructors, this report focuses on one participant, Disha, and her use of a questioning technique called hypophora. At the beginning of the observations, 25% of her questions were hypophora. After video-case based activities during weekly coordination meetings, her use of hypophora decreased to about 10% of questions. Although Disha rejected the idea that her teaching had changed in any way, she acknowledged that she began “breaking things into smaller pieces” to help students understand.
2017
San Diego, California
In an effort to understand ways students approach constructing homomorphisms and isomorphisms between groups, six undergraduate math and engineering students in a lecture- based introductory abstract algebra course were interviewed. These students experienced varied success in creating isomorphisms and homomorphisms, which allowed both successful techniques for map creation and stumbling blocks to map creation to emerge from the data. Additionally, a genetic decomposition for homomorphism is outlined and students’ mental constructions of both homomorphism and isomorphism are discussed. Finally, students’ conceptual metaphors for homomorphic and isomorphic mappings are examined. Combining these three analyses paints a picture of the interaction between students’ knowledge of properties of groups and mappings and their flexibility in creating mental images while interpreting those properties.
2017
San Diego, California
Only recently ‘abstraction from objects’ has attracted attention in the literature as a form of abstraction that has the potential to take account of the complexity of students’ knowing and learning processes compatible with their strategy of giving meaning. This paper draws attention to several emerging insights from the evolving framework of structural abstraction in students’ knowing and learning of the limit concept of a sequence. Particular ideas are accentuated that we need to understand from a theoretical point of view since they reveal a new way of understanding knowing and learning advanced mathematical concepts.
2017
San Diego, California
This study investigates Calculus, Transition-to-Proof, and Advanced Calculus students’ meanings for quantifiers in conditional statements involving multiple quantifiers. Three students from each course participated in clinical interviews. Students were presented with the Intermediate Value Theorem (IVT) and three other statements whose logical structure was similar to the IVT except for the order of both the quantifiers and their attached variables. The results reveal that Advanced Calculus and Transition-to-Proof students made distinctions between the different statements more often than Calculus students. Several student meanings for quantification were found to be necessary for making distinctions between each of the four statements. We also address student quantifications for hidden quantifiers in the statements.
2017
San Diego, California
We report a qualitative analysis of 14 undergraduate students’ experience in a semester long introduction to proof course. Half were mathematics majors. Our research aims to characterize, conceptually and empirically, students’ transition from a focus on computation to proof in mathematics. Our analysis focused on how students saw the course as different from prior courses, whether it required new or different learning activity, how they described their work in proof-writing, and how they saw their confidence and success in the course. This approach— targeting students’ overall experience of the course—differs from prior research that has tracked students’ challenges, focused on their work on specific proof problems, and explored how to support and improve their work (e.g., Selden & Selden, 2003). Our work has promise for informing the design of transition to proof courses and how those courses are organized and taught.
2017
San Diego, California
Eigentheory is a conceptually complex idea whose application is widespread in mathematics and beyond. Herein we describe the development and use of an extended multiple choice assessment that gives us further insight into the ways students think about and understand eigenvectors, eigenvalues, and their related concepts.
2017
San Diego, California
In this report we share analysis regarding students’ meta-representational competence (MRC) that is expressed as they engage in solving quantum mechanics problems that involve linear algebra concepts. The particular characteristic of MRC that is the focus of this analysis is students’ critiquing and comparing the adequacy of representations, specifically matrix notation and Dirac notation, and judging their suitability for various tasks (diSessa, 2004). With data from semi-structured individual interviews, we created categories of types of MRC elicited during students’ work on an expectation value problem. We provide detail on two students who serve as paradigmatic examples of a student’s power and flexibility in thinking in and using different notation systems. This work lends credence to and inspires our preliminary conjecture that strong meta-representational competence (MRC) is necessary not only to be fluent and proficient in the mathematics involved in solving quantum mechanics problems but also to develop a robust understanding of the quantum mechanics content.
2017
San Diego, California
We examine multiple data sources to assess the current progress of implementing corequisite remediation and multiple math pathways in the state of Oklahoma. We begin by analyzing trends in national reform efforts and contrasting them with the status of current challenges and efforts in Oklahoma. We then present preliminary data from pilot sections of a corequisite College Algebra course and a new math pathway for degrees that require significant quantitative literacy but do not require engineering calculus. Finally, we consider statewide data on student course- taking patterns, degree requirements, and existing institutional efforts that will inform upcoming state-level decisions on these reforms.
2017
San Diego, California
Research and surveys continue to perpetuate deficit narratives about women of color, particularly regarding their participation in and contribution to mathematics. Following the broader call for more research concerning STEM learning experiences of women of color, this study focuses on the sense making of eight women of color regarding their understanding of basis in linear algebra. We documented diverse ways that these women creatively explained the concept of basis using intuitive ideas from their everyday lives. These examples revealed important nuances and aspects of understanding of basis that are rarely discussed in instruction. These students’ ideas can also serve as potentially productive avenues to access the topic. Our results also challenge the existing broader narrative about academic underachievement of women of color in mathematics.
2017
San Diego, California
A variety of computerized learning platforms exist. In mathematics, most include sets of problems to complete. Feedback to users ranges from a single word like “Correct!” to offers of hints and partially- to fully-worked examples. Behind-the-scenes design of such systems also varies – from static dictionaries of problems to responsive programming that adapts assignments to users’ demonstrated skills within the computerized environment. This report presents background on digital learning contexts and early results of a mixed-methods study that included a cluster randomized controlled trial design. The study was in community college algebra classes where the intervention was a particular type of web-based activity and testing system.
2017
San Diego, California
In this paper we explore the ways in which mathematicians talk about explanation in their research papers. We analyze the use of the words explain/explanation (and various related words) in a large corpus of text containing research papers in both mathematics and physical sciences. We found that mathematicians do not frequently use this family of words and that their use is considerably more prevalent in physics papers than in mathematics papers. In particular, we found that physicists talk about explaining why disproportionately more often than mathematicians. We discuss some possible accounts for these differences.
2017
San Diego, California
In Where Mathematics Comes From, Lakoff and Núñez (2001) describe how the notions of infinity and limit can be constructed through metaphorical extensions of embodied experiences. This paper will critique their historical and psychological analysis, revealing an unresolved tension between a simplified, geometric “approaching” conception and the arithmetization of calculus by Weierstrass. A proposal of how to rectify this conflict through acknowledging how novices can metaphorically tie these concepts together is discussed.
2017
San Diego, California
We document Alice’s progression with proof-writing over two semesters. We analyzed videotapes of her one-on-one sessions working through the course notes for our inquiry-based transition-to-proof course. Our theoretical perspective informed our work and includes the view that proof construction is a sequence of mental and physical, actions. It also includes the use of proof frameworks as a means of getting started. Alice’s early reluctance to use proof frameworks, after an initial introduction to them, is documented, as well as her subsequent acceptance of and proficiency with them by the end of the real analysis section of the course notes, along with a sense of self-efficacy. However, during the second semester, upon first encountering semigroups, with which she had no prior experience, her proof writing deteriorated, as she coped with understanding the new concepts. But later, she began using proof frameworks again and regained a sense of self-efficacy.
2017
San Diego, California
One common approach to assessing mathematical knowledge for teaching (MKT) is designing items to measure individual subdomains of MKT as specified by a theoretical framework, with factor analyses confirming (or disconfirming) hypothesized subdomains. We interpret this approach as adhering to a “compartmentalized” view of MKT, as opposed to a “connected” view of MKT. We argue in this paper that a compartmentalized view of MKT is embedded in the ways that frameworks are represented, discussed, and used in the field, but that this view of MKT may unintentionally undermine understanding of MKT, and in turn, how to measure and cultivate it in teachers. Using an analysis of nine items previously shown to assess MKT, we illustrate that the tendency for practice-based items to capture multiple subdomains is a common issue across frameworks, and perhaps one that is a necessary result of their design.
2017
San Diego, California
This theory-based report gives evidence and builds a conceptual framework for a construct called “mathematical knowledge for teaching future teachers” (MKT-FT). Mathematics teacher educators construct MKT-FT as they teach courses for pre-service teachers. Connections to mathematical knowledge for teaching (MKT) are discussed, with an emphasis on the complex relationships among aspects of pedagogical content knowledge in MKT-FT and MKT.
2017
San Diego, California
Geometry is the subject where U.S. students are weakest on international assessments, but college geometry is an area of proof that is understudied. Since geometry is secondary students’ only exposure to proof, it is vital our secondary teachers can prove effectively in this content area. The purpose of this case study, drawn from a larger project, was to understand how, if at all, pre-service teachers’ proof schemes became more axiomatic throughout a one-semester inquiry-based college geometry course. Participants in this study, Alexis and Lindsey, were pre-service teachers enrolled in an inquiry based college geometry course. Although Alexis and Lindsey had differing experience with proof at the start of the course, the change to revise their proofs and discuss problems with their peers helped both students advance to more axiomatic geometric thinking.
2017
San Diego, California
Proof is central to the curriculum for undergraduate mathematics majors. Despite transition-to- proof courses designed to facilitate the transition from computation-based mathematics to proof- based mathematics, students continue to struggle with all aspects of mathematical proof. In particular, research suggests that proof by contradiction is an especially difficult proof method for students to construct and comprehend and yet, there are no satisfactory instructional models for how to teach the method. The purpose of this paper is to discuss preliminary results of a teaching experiment on student comprehension of proof by contradiction within a transition-to- proof course. Grounded in APOS Theory, this paper will illustrate that a student’s conception of mathematical logic, and quantification in particular, plays an important role in their comprehension of proof by contradiction.
2017
San Diego, California
Research and surveys continue to document the underrepresentation of women of color (WOC) in mathematics. Historically, their achievement in mathematics has been framed in a deficit way. Following the broader call for more research concerning WOC’s learning experiences in STEM, we interviewed eight WOC about their understanding of basis in linear algebra. We documented diverse ways that these women creatively explained the concept of basis using intuitive ideas from their everyday lives. These examples revealed important nuances and aspects of understanding of basis that are rarely discussed in instruction. These students’ ideas can also serve as potentially productive avenues to access the topic. Our results also challenge the existing broader narrative about the underachievement of women of color in mathematics.
2017
San Diego, California
Corpus linguists attempt to understand language by statistically analyzing large collections of text, known as corpora. We describe the creation of three corpora designed to enable the study of expert and learner mathematical language. Our corpora were formed by collecting and processing three different genres of mathematical texts: mathematical research papers, undergraduate-level textbooks, and undergraduate dissertations. We pay particular attention to the method by which our corpora were created, and present a mechanism by which LaTeX source files can be easily converted to a form suitable for use with corpus analysis software packages. We then compare these three different types of mathematical texts by analyzing their word frequency distributions. We find that undergraduate students write in remarkably similar ways to textbook authors, but that research papers are substantially different. These differences are discussed.
2017
San Diego, California
Many universities have begun to coordinate their introductory mathematics courses to handle multiple sections of the same course, a situation necessitated by the large numbers of students taking precalculus and calculus. Robust coordination systems consist of two major elements: uniform course elements (e.g., common text; exams) and regular instructor meetings. These regular meetings may turn calculus instruction into a joint enterprise, potentially engendering a community of practice. Of particular importance are those who act as leaders (formally and/or informally) within these coordination systems – these people have the potential to influence and “nudge” instructors towards improving their practice (Rasmussen & Ellis, 2015). This study combines case study findings with social network data to investigate instructional leaders at five diverse institutions, considering both formal and informal coordination phenomena, and hypothesize about their potential to influence practice in their departments.
2017
San Diego, California
As a result of the increased focus on data literacy and data science across the world, there has been a large demand for teacher preparation in statistics. Project-SET constructed two hypothetical learning trajectories for teacher learning and subsequently used the hypothetical learning trajectories to structure a professional development curriculum. We illustrate how the utilization of learning trajectories to design professional development allowed participating teachers to develop several aspects of Statistics Knowledge for Teaching (Groth, 2013).
2017
San Diego, California
In this report we present a taxonomy of mathematics graduate student teaching assistant (GTA) professional development (PD) programs. This taxonomy is based off of the characterization of GTA PD programs from 120 mathematics departments, and is informed by the framework developed by Ellis (2015) based on case studies of four GTA PD programs. A cluster analysis revealed nine distinct models of GTA PD within the 120 programs. These nine models vary with respect to the amount of interaction the GTAs have through the PD, the amount of activities involved in the PD, and the level of feedback given to GTAs involved with the PD. We present a characterization of one of the nine models using Ellis’s framework.
2017
San Diego, California
The purpose of the reported study is to explore students’ reasoning about the “within argument contradictions” that arise from logically degenerate cases by analyzing the problematics noticed in students’ proof scripts. The work proposes a framework for students’ noticed proof problematics and explores the viability of the proof script methodology as a mechanism for identifying difficulties experienced by students but unseen by experts. In the case of logically degenerate cases, findings indicate students held conceptions of proofs by cases that inhibited students’ reasoning about the encountered contradictions.
2017
San Diego, California
In this study, we conducted a teaching experiment with two students to investigate the development of a generalized concept of inverse in abstract algebra. In particular, we document the stages through which the students’ reasoning progressed, initiating with an understanding of the additive inverse of an element as the result of a procedure applied to that element, and concluding with a generalized understanding of inverse that was broad enough to identify instances of inverses in various and unfamiliar algebraic structures. Of critical importance was the development of and coordination with a corresponding concept of identity.
2017
San Diego, California
A number of institutions with mathematics programs offer introduction to proof courses in order to ease mathematics students’ transition from primarily calculation-based courses to proof- centered courses. However, unlike most tertiary mathematics courses, whose mathematical content is directly implied by their course titles, introduction to proof courses may vary in terms of the mathematics content discussed. In this study we document the variation in content of introduction to proof courses. This is achieved by examining recent syllabi and other relevant course documents from introduction to proof courses at 179 R1/R2 universities across the United States. The various types of content used in these courses are discussed. We describe the 15 categories of ITP courses that emerged from the course information we collected and offer our categories as a framework for classifying ITP courses or students in future studies.
2017
San Diego, California
The purpose of this study is to examine the role of visual reasoning while students evaluate complex mathematical statements about real-valued functions. We conducted clinical interviews with nine undergraduate students from mathematics courses at different levels. In the interviews, we asked these students to evaluate several mathematical statements alone and then using various graphs. In this paper, we focus on the cases of two students who had completed Advanced Calculus to highlight the contribution of their visual reasoning about several graphs. We found that student’s graphical interpretation of “between” in these statements affected their evaluation of the statements. Even at advanced levels, students’ visual cues dominated their reasoning about the statements. Our findings indicate that students’ visual reasoning contributes to their evaluation of mathematical statements and helps to account for differences between students’ meanings of statements.
2017
San Diego, California
In mathematical research as well as pedagogy, mathematicians rely on proofs to convey mathematical knowledge. Both mathematicians and mathematics educators have argued that a proof is more valuable to students when it explains why a theorem is true. In this contributed report, I discuss attributes of explanatory proofs that eleven doctoral students in mathematics described. Doctoral students in this study interpreted the nature of mathematical explanation in the context of a proof in a wide range of ways. In particular, these participants expressed that they are more likely to consider a proof more explanatory when it succeeds in providing (a) insight into the derivation of certain formulas, (b) intuition as to why the theorem is true, or (c) insight into how the author or the reader could have discovered the proof in practice.
2017
San Diego, California
In this paper I explore eleven undergraduate students’ comprehension of a proof taken from an undergraduate abstract algebra course. My interpretation of what it means to understand a proof is based on a proof comprehension model developed by Mejia-Ramos, et al. (2012). This study in particular examines the extent to which undergraduate students are able to summarize a proof using the proof’s higher-level ideas. Additionally, eleven doctoral students in mathematics were asked to provide a summary of the same proof that the undergraduate students received. Undergraduates’ holistic comprehension of the proof was then analyzed in light of summaries that the doctoral students provided. The main finding of the study is that undergraduates’ comprehension of the proof was overall inadequate—notably, they demonstrated limited skills in summarizing a proof via the proof’s key ideas. Moreover, undergraduates failed to recognize the scope of the method used in the proof.
2017
San Diego, California
A function is defined as a mapping from one nonempty set (the domain) to another nonempty set (the co-domain or range) such that each element of the domain maps to exactly one element of the range. Algebra curricula typically include classification tasks in which students determine if a relation violates the univalence criterion – the condition that each element in the domain corresponds to exactly one element of the range. This paper provides a longitudinal case study of how one student generalised the univalence criterion from single- to multivariable functions. For f(x), Kyle primarily thought of univalence in terms of the vertical line test and the variables x and y. He generalised univalence for the multivariable function f(x,y) by thinking about input, output, independence, and dependence. Kyle’s story provides an example of how a student might generalise facets of the function concept in normatively correct ways.
2017
San Diego, California
The Stimulated Construction of Narratives about Interactions (SCNI) technique for data collection, introduced in this paper, enables robust investigations of small-group learning at college. The SCNI technique consists of promptly soliciting participants’ perspectives on their recent joint activity using video records thereof. Thus the SCNI technique creates a space to network the narrative discourses that shape how participants understand their world, and the pragmatic forces that shape participants’ interactions in a practice. Data reported in this paper are collected from a number theory class comprised of ethnically diverse students. In this paper, I will report three cases to illustrate the advantages of SCNI data over data collected by video records and unmediated interviews in elucidating, nuancing, and expounding what matters for group work. Through these cases, I will use three different analyses appropriate for SCNI data. Limitations and recommendations for efficient conduct of SCNI are discussed as well.
2017
San Diego, California
Two representation registers are described that support student reasoning with definite integral notation: adding up pieces (AUP) and multiplicatively-based summation (MBS). These registers were developed in a Calculus I class that used an informal infinitesimals approach, through which differentials like dx directly represent infinitesimal quantities rather than serving as notational finesses or vestiges. Student reasoning reveals how the AUP register supports modeling with integral notation and how the MBS register supports sense-making with and evaluation of integrals.
2017
San Diego, California
Counting problems provide rich mathematical content and a variety of applications for students, motivating investigation into the difficulties students face while counting. In particular, an important result supported by previous quantitative and qualitative evidence is that listing may be an effective strategy for combating some student struggles in counting, particularly since it draws explicit attention to outcome structure. However, anecdotal experience has shown that students can resist listing and feel that it is tedious and not worth the effort. To investigate whether these negative mindsets exist outside these anecdotes, task-based interviews were conducted targeting student attitudes toward listing and their success in using listing to solve counting problems. Contrary to the anecdotal evidence, the students in the study expressed that they felt listing is a worthwhile activity, but their work on counting problems suggest that they would benefit from more explicit support relating to listing in their discrete mathematics classes.
2017
San Diego, California
Researchers have described the importance of seeing a graph as an emergent trace of how two quantities’ values vary simultaneously. Researchers have also identified the many difficulties students face when constructing this conceptualization of graphs. In this paper I explore the role of two didactic objects on a student’s conceptualization of graphs. In particular, I examine how a student’s interactions with these didactic objects supported her in making key differentiations that enabled her to conceptualize a graph as emerging from simultaneously tracking two quantities’ varying values. My findings revealed that a student must differentiate a place on a function’s graph from the value of the function’s output. Also, the student must distinguish tracking a point in the plane from creating the point by simultaneously attending to the variation of two quantities.
2017
San Diego, California
This study investigates differences in mathematical self-efficacy and outcome expectations of 3107 incoming students enrolled in introductory level mathematics or statistics courses at a land grant university in the Midwest. Students were grouped by discipline (STEM (Science, Technology, Engineering and Mathematics), Social Sciences and Arts & Humanities) and by gender within each discipline. All students enrolled in an introductory mathematics or statistics course during their first semester at the institution were surveyed about their perceived mathematical self-efficacy and outcome expectations at the beginning of that semester. Our results suggest that discipline specific differences are dependent on the definition of STEM majors, namely distinguishing between math intensive and non-math intensive STEM majors. After accounting for this distinction gender differences in mathematical self-efficacy and outcome expectations disappear.
2017
San Diego, California
This paper highlights the data from a one-semester course with pre-service teachers in an ongoing study of their conceptions of area at a public university in the western United States. Their meanings of area and area units, both standard and non-standard, were explored throughout the semester. Analysis of our interviews with these pre-service teachers about their responses to area tasks allowed us to uncover three cognitive conflicts in conceptualizing area, namely: 1) how can non-square units be square(d)?, 2) how can we find the area of a shape when the area unit is neither square nor polygonal?, 3) how can we use square or polygonal area units to measure the area of a shape with curved boundaries? The work of one representative case study is reported here. This work will help educators develop tasks to initiate these cognitive conflicts for improved conceptualizations of area.
2017
San Diego, California
The Common Core State Standards recommend students to decontextualize word problems using symbols and contextualize symbols by defining the meaning of values. We observed pre-service teachers’ difficulties with contextualizing symbols in word problems; hence, we incorporated supplementary word problems’ modeling instructions for an arithmetic course with pre-service teachers. After six weeks of instruction, on a midterm exam, our pre-service teachers started using symbols to present arithmetic word problems. However, many of them still could not clearly define the meaning of the symbols they used. After completing the program, students demonstrated improvement in their reasoning with symbols. We believe difficulties with defining symbols are connected to weaknesses with active scientific vocabulary in terms of measurable attributes. Therefore, we propose mathematics courses for prospective teachers to accentuate scientific vocabulary regarding measurable attributes.
2017
San Diego, California
In this paper we report on a study of assessment-based oral presentation tasks in a statistics course at a public university in the United States. We examine student attitudes towards using oral presentation tasks in learning statistics and their disposition towards statistics as well as their knowledge of the statistical concepts. Our results suggest that use of oral presentation improves students’ mastery of statistical concepts and their disposition towards statistics. Moreover, responses to the anonymous course evaluation questionnaire provide insights on the benefits of using oral presentation tasks in statistics courses for students.
2017
San Diego, California
A variety of computerized interactive learning platforms exist. Most include instructional supports in the form of problem sets. Feedback to users ranges from a single word like “Correct!” to offers of hints and partially- to fully-worked examples. Behind-the-scenes design of systems varies as well – from static dictionaries of problems to “intelligent” and responsive programming that adapts assignments to users’ demonstrated skills within the computerized environment. This report presents background on digital learning contexts and early results of a cluster-randomized controlled trial study in community college elementary algebra classes where the intervention was a particular type of web-based activity and testing system.
2017
San Diego, California
While studies continually show benefits of active learning strategies like inquiry-based learning (IBL), it is difficult to get faculty to adopt these methods. Particularly challenging is the third and final stage in Paulsen and Feldman’s (1995) model, ‘refreezing,’ when instructors use feedback and support to decide whether to continue with the instructional changes they have made or return to their previous methods. In this paper, we show how a workshop to teach college mathematics instructors to implement IBL used both online and in-person communities to help provide the ongoing feedback and support necessary for ‘refreezing.’ We offer lessons for how to increase the relevance of and participation in online support communities. We also use an innovative analytical approach, Social Network Analysis, to understand the ongoing processes of how e-mail exchanges provide feedback and both intellectual and emotional support to workshop participants.
2017
San Diego, California
This report explores pre-service teachers’ proficiency with concepts of transformational geometry at the end of a semester-long advanced geometry course. During the course, the instructor presented transformational geometry content, including congruence proofs, in an attempt to align with the Common Core State Standards for Mathematics. At the end of the course, the students, all pre-service teachers, appeared to mix ideas from the traditional approach involving triangle congruence criteria (SAS, ASA, SSS, AAS) and transformational approaches, and struggled with conceiving of transformation functions as objects. These difficulties appear to compound in their proof-writing attempts such that after citing appropriate transformational geometry ideas, such as the angle- or distance-preservation property, they would then supplement with congruence-based approaches in order to finish the proof. This has implications, especially for professional development, as this is the final mathematics class that these preservice teachers will take concerning transformational geometry prior to beginning their classroom instruction.
2017
San Diego, California
In this paper, we discuss two experts’ reasoning abilities when tasked with drawing a graph that relates two varying quantities. We present evidence that in some cases, these experts had constructed and coordinated the amounts of change of the quantities (while interpreting and constructing graphs). By comparing each experts’ activities and corroborating previous researchers’ findings, we argue that constructing a multiplicative object is critical to conceiving a graph and situation as constituted by covarying quantities. We identify particular complexities involved in the development of covariational reasoning including the conceptualization, coordination, and referent accumulation of the amounts of change of two quantities.
2017
San Diego, California
This study expands on research happening in multivariate calculus education to an exploration of student understanding of line and vector integrals. We describe how students interpreted these types of integrals, including the various symbols in their expressions and their relationships to each other. We found that while students were able to associate mathematical objects with the individual symbols in the integral expressions, the main issues came in trying to coordinate these objects into a comprehensive whole for the entire integral expression. While the literature has discussed connections between the integrand, differential, and integral symbol, we also found that connecting the domain to the other parts of the integral expression was difficult as well. Thus, to give more attention to the domain in addition to these other parts, we have incorporated it into a general “domain-chop-evaluate-add” framework for reading integrals.
2017
San Diego, California
Much of the calculus education research on student understanding of integrals has been separated into definite-integral-focused studies and indefinite-integral-focused studies. This means that research may not be capturing how students might see these two types of integrals in relation to one another, as opposed to in isolation of each other. This study examines whether students see these two types of integrals as representing the same basic concept, distinct concepts, or as sharing some concepts while diverging in others. The results show that a large majority of students ascribe the exact same underlying conceptions to both types of integrals, even describing the indefinite integral as representing the area under a curve also. We relate what aspects of each type of conception students saw as common to both types of integrals, and what features of the conception the students saw as different between them.
2017
San Diego, California
Faculty in undergraduate mathematics departments are currently involved in making changes to their instruction, particularly by introducing different modes of student-centered type instruction. In this paper, we analyze a situation where faculty are involved in online collaboration using video of their own classrooms. We found that showing the video during the online work groups promotes more discussion of pedagogy rooted in instructional components than instructors watching the videos alone before. Pedagogy and students’ mathematics also become important discussion points that encouraged and supported the instructors. Providing instructional components as a frame proved to be successful in supporting the video discussions as they stay centered on instruction.
2017
San Diego, California
Reforming the way undergraduate math is taught has been the target of significant research efforts for decades; however, lecture remains the predominant form of instruction. While interest has been primarily focused on entry-level courses in order to recruit and retain STEM- intending students, quality instruction in upper division courses is also important. In a national survey of abstract algebra instructors, we investigated typical teaching practices, beliefs, and constraints that influence pedagogical decisions, and similarities/differences between those who do/not lecture. Of particular interest was exploring whether instructors at Bachelor’s-granting institutions have markedly different circumstances than their counterparts at Master’s- and Doctoral-granting institutions and the effect (if any) this has on their pedagogical decisions.
2017
San Diego, California
This proposal discusses the extent to which mathematicians agree amongst themselves with regard to what are some of the linguistic conventions of mathematical proof writing. Data from a survey of 128 mathematicians are used to address this question. Participants were asked whether various excerpts highlighted in four partial proofs were unconventional in each of three different contexts: how proofs appear in undergraduate mathematics textbooks, what instructors write on the blackboard in undergraduate mathematics courses, and how students write proofs in these courses. These data point to a lack of agreement among mathematicians on the linguistic expectations of the proofs written by their students.
2017
San Diego, California
In an effort to better understand students’ understanding of the multiplication principle, which is a fundamental aspect of combinatorial enumeration, we had two undergraduate students engage in reinvention of a statement of the principle during an eight-session teaching experiment. In this presentation, we report on the students’ unexpected attention to the order in which they complete stages of counting process in a counting problem. We suggest that an early experience with a particular problem prompted them to think about order, and this way of thinking persisted throughout the experiment. The students’ reasoning about order sheds light on ways in which students may think about order and about the nature of multiplication in counting. We conclude with potential implications and directions for further research.
2017
San Diego, California
We report on a mathematician’s perceptions and awarenesses related to incorporating problem-based activities requiring computational thinking into an upper level undergraduate mathematics course. Computational thinking is understood as the thinking, strategies, and approaches for problem solving that parallel the design of computational algorithms which can be followed and executed by a computer. Data from this case study is qualitative in nature, and seeks to present an in-depth account of one professor’s experiences developing and teaching computational thinking in and for mathematics. Analyses highlight the similarities and differences amongst the values and opportunities perceived for computational thinking versus other more ubiquitous mathematical approaches, as well as the perceived tensions and challenges in trying to foster such values and opportunities.
2017
San Diego, California
In this paper we explore the ways in which mathematicians talk about explanation in their research papers. We analyze the use of the words explain/explanation (and various related words) in a large corpus of text containing research papers in both mathematics and physical sciences. We found that mathematicians do not frequently use this family of words and that their use is considerably more prevalent in physics papers than in mathematics papers. In particular, we found that physicists talk about explaining why disproportionately more often than mathematicians. We discuss some possible accounts for these differences.
2017
San Diego, California
How do nine instructors teaching a linear algebra course at a research university manage tensions that emerge because of the requirement of teaching the course with an Inquiry-Based Learning approach within a coordinated system? Using Herbst’s practical rationality framework (Chazan, Herbst, & Clark, 2016; Herbst & Chazan, 2011) we identify features of the course organization that contributed to tensions between professional obligations that were resolved via the production of worksheets that teachers gave to the students. We noted differences in how these tensions were handled, and provide some evidence that such differences might be related to the research orientation the instructors brought and to their status in the institution. We formulate some hypotheses that can shed light on how to assist in changing post-secondary instructional practices.
2017
San Diego, California
Exploring students’ conceptions of sameness is an avenue for exploring their understandings of the objects being compared. More specifically, finding what students think it means for functions to be identical can help us figure out what students think it means for something to be a function, since identity within a category (in this case the category of function) is inextricably tied to the defining aspect of that category. This paper has three primary aims: to illustrate the importance of using students’ assessments of sameness as a means to discover their concept images, to describe a particular student’s concept image of function and of function sameness, and to suggest that the math education research community develop a more refined understanding of a “process”(cf., Breidenbach, Dubinsky, Hawks, & Nichols, 1992) conception of function.
2017
San Diego, California
The purpose of this study was to investigate students’ quantitative reasoning when solving a multivariable problem in a revenue maximization context. We conducted task-based interviews with 12 pairs of business calculus students. Analysis of verbal responses and work written by the students revealed that in reasoning about the relationships among the quantities (sales, discount, and total revenue) in the problem, nearly all the pairs of students created new quantities. The creation of these quantities helped the students to reason about the effect of the discount on sales and total revenue. An important finding of this study is that the students took different approaches to the meaning of the discount and only five pairs of the students interpreted the discount as intended in the design of the problem. Directions for future research are discussed.
2017
San Diego, California
In this study, I examined 14 preservice secondary teachers’ abilities to transfer graphical to algebraic representations of functions. The analysis showed that the vast majority of the participants had problems in noticing critical behaviors of function graphs and in using them to construct algebraic forms. About half or fewer of the participants noticed qualities such as x- intercepts, vertical asymptotes, slant asymptotes, and concavity/extrema, with only a few of them successfully using such qualities in constructing algebraic forms. Only a few noticed and used qualities such as horizontal asymptotes, point discontinuities, domain, and end behaviors in constructing algebraic forms. It is advisable that the teaching of the function concept incorporate transformational activities beyond algebraic to graphical transformations and focus more on the critical characteristics of functions.
2017
San Diego, California
This study reports three preservice secondary teachers’ abilities and tendencies to use representations in problem solving as well as their abilities to use realistic tasks after taking mathematics content and methods courses that emphasized the roles of representations and realistic tasks. Qualitative analyses showed that the preservice teachers developed beliefs that representations and realistic tasks are important components of secondary education and used motivational tasks in their instruction. However, they used the tasks mainly as the application of learned facts rather than as the departure of students’ construction of mathematical ideas. They also showed tendencies to use algebraic approaches in problem solving for grade 5-12 level tasks and had difficulties connecting algebraic and geometric representations when solving high school level algebra problems.
2017
San Diego, California
Little is known about preservice elementary teachers’ understandings of greatest common factor (GCF) or how they relate to their understandings of least common multiple (LCM). As part of a larger case study in which an emergent perspective (Cobb & Yackel, 1996) was used to investigate preservice elementary teachers’ understandings of topics in number theory, task- based interviews elicited participants’ conceptions about modeling GCF and LCM using manipulatives, pictures, and story problems and the procedures for finding GCF and LCM using prime factorizations. Additional classroom data served to support findings. Participants held stronger understandings of modeling LCM than they did with modeling GCF. In contrast, participants’ understandings of the procedure for finding GCF were far more robust than their understandings of how to find LCM.
2017
San Diego, California
There has been a substantial increase in mathematics education research in how proof-oriented university mathematics courses are traditionally taught. In this paper, we focus on the questions that lecturers pose to students. Specifically, we audio-recorded 11 proof-oriented mathematics lecturers and analyzed all of the questions they asked their students. We categorized each of the 1,031 questions according to a coding system we describe as well as identified wait time and subsequent speaker. We describe trends across all 11 lecturers, highlighting the limited opportunities students had to engage in important mathematical practices, and identify variances between how different lecturers used questions. We present qualitative data highlighting common and uncommon questioning techniques and conclude with a discussion of our results.
2017
San Diego, California
In this study, I explored the use of a worked-examples-based proof-writing framework as a pedagogical tool to improve undergraduate students’ ability to construct proofs. Over the course of three months, I ran a series of three workshops with five undergraduate students who had no prior experience with formal mathematical proof. In each workshop, participants worked through worksheets containing completed worked examples of mathematical proofs, followed by partially completed worked examples of proofs (to be completed by the participants), and, lastly, exercises. I collected and coded participants’ written work and reflections and explored changes in student proof-writing across workshop sessions. In this paper, I describe themes across student work and provide qualitative data supporting the benefits of incorporating the use of such a worked-examples-based proof-writing framework when introducing students to mathematical proof.
2017
San Diego, California
This study explored how one Korean and one U.S. calculus class defined the word “derivative” as a point-specific object through the limit process on the difference quotient, and as a function on its domain. The analysis using Commognitive approach showed that both class used similar visual mediators for the limit process/object, but addressed different components of the definitions; Discussion of the derivative as a function before it was defined were frequently found in the U.S. class but rarely found in the Korean class; Words for the derivative at a point, and words for the derivative as a function explicitly differed in the Korean class compared to the U.S. class; and the derivative was first defined as a function through correspondence between x-value and the derivative value in Korean class, but through expansion of x values from a number to variable and corresponding changes in the U.S. class.
2017
San Diego, California
This study evaluated the outcomes of an intervention focused on developing mathematics graduate teaching assistants’ (GTAs’) skills of noticing and effectively responding to instances of student mathematical thinking that have significant potential to further students’ learning. Four GTAs participated in a semester-long intervention that included individual analysis and group discussion of video of undergraduate mathematics lessons. The MOST Analytic Framework (Stockero, Peterson, Leatham, & Van Zoest, 2014) was introduced to aid in these activities. The GTAs also completed a pre- and post-interview to document their real time noticing and an assessment of common content knowledge. Results indicate that the intervention was successful in improving the GTAs’ noticing skills in a variety of ways and in their ability to propose student-centered responses.
2017
San Diego, California
In this study, we explore the norms by which students and undergraduate mentors in a summer mathematics program evaluate proofs of theorems in number theory. By utilizing cognitive interviews during which students and mentors evaluate number theory proofs written by a hypothetical student, we find that for students as well as mentors, “rigor” is a dimension of mathematical acceptability of proofs distinct from, though related to, proof validity. Additionally, we find that both students and mentors frequently adhere to strict unwritten norms that govern how they believe proofs should be constructed and presented, and that these norms may be more rigid than the intended proof-writing norms of the mathematicians who teach in the summer program. This study suggests some potential challenges associated with the growing practice of asking undergraduate student graders to evaluate proofs written by students in introduction-to- proof courses.
2017
San Diego, California
In an introductory linear algebra course, students are expected to learn a plethora of new concepts as well as how these concepts are connected to one another. Learning these connections can be quite challenging for students due to the vast number of connections and student inexperience with mathematical logic. The study reported here consisted of an investigation into how inquiry-oriented teaching methods could be employed in an attempt to create opportunities for students to develop mathematical connections in an introductory linear algebra course.
2017
San Diego, California
In this paper I re-analyze the transcripts from Smith (2012) to investigate silence in mathematicians’ collaborative work. I provide an existence proof that silence, at times, forms a significant aspect of mathematicians’ embodied work. Based off a discussion of the nature of embodied interaction, the paper concludes that it is likely that silence forms a significant aspect of mathematicians’ collaborative work, more generally, both in discovering new mathematics, and in ordering the mathematicians together toward the task of discovering new mathematics. Because this use of silence is different from that of everyday conversation, this raises important pedagogical questions regarding students’ apprenticeship into the mathematics profession.
2017
San Diego, California
Graduate Student Instructor (GSI) professional development addresses an urgent need to improve STEM retention. This paper focuses on a semester-long professional learning community in which six mathematics GSIs engaged in regular cycles of peer observation, feedback, and reflection. In contrast to most GSI development work, this approach emphasized that GSIs give, not just receive, peer feedback. Analyses of post-semester interviews indicated that all GSIs enhanced their noticing of students. Moreover, insight into peer feedback was developed along three dimensions: (1) the importance of being an objective observer, (2) the impact of working with equal-status peers, and (3) the value of critical feedback.
2017
San Diego, California
Given its applications in computing, coding, and cryptography, the Chinese Remainder Theorem is a worthwhile, accessible, and unexplored area of number theory. The purpose of this qualitative case study was to investigate strategies and reasoning that students exhibited while solving problems chosen to elicit thinking in elementary number theory topics related to the Chinese Remainder Theorem. We interviewed pairs of students from three different courses in order to investigate the similarities and differences that may occur as a result of varying mathematical backgrounds and partner dynamics. We identified a range of strategies including manipulating final digits, listing multiples while accounting for remainders, and implementing divisibility rules. This paper presents a portion of our findings discussing strategies for two of our three cases on several tasks from our interviews.
2017
San Diego, California
In an effort to understand ways students approach constructing homomorphisms and isomorphisms between groups, six undergraduate math and engineering students in a lecture- based introductory abstract algebra course were interviewed. These students experienced varied success in creating isomorphisms and homomorphisms, which allowed both successful techniques for map creation and stumbling blocks to map creation to emerge from the data. Some successful techniques for determining if groups were isomorphic included checking the orders of the groups, looking for invertible maps between groups, and determining the identity element and orders of elements of each group. Successful strategies for approaching the creation of homomorphisms included checking if the groups were isomorphic, seeing if a proposed map would preserve closure, and using strategic trial and error. Stumbling blocks included the inappropriate use of definitions, an inability to interpret definitions, and misunderstanding the distinction between the names and roles of elements in different groups.
2017
San Diego, California
What factors (in terms of the student) contribute to success in college calculus, and what are the relationships between and relative importance of these factors? This study addresses these questions by building on the Academic Performance Determinants Model (Credé and Kuncel, 2008). A new model called the Success Factor Model for Calculus was developed using semi- structured and task-based interviews with fourteen first-semester college calculus students. The data suggests that creative mathematical reasoning and knowing-why are not required for success on college calculus tests. Alternatively, motivation is a determining factor in success in that students can perform well on exams by being motivated to know how to solve specific types of problems. Motivation is decreased by some course-specific factors, such as lack of structure and accountability, and its effect on success is decreased sometimes by a lack of study skills and habits.
2017
San Diego, California
With this research, we seek to find theoretical constructs that correlate with participants’ neural activity that occurs as they are presented slides of mathematical proofs. We first asked three graduate student participants to complete two graduate level proofs (one each of abstract algebra and real analysis) using a LiveScribe pen. We then generated slides of their written work and researcher-generated proofs that we used during electroencephalography (EEG) trials. Having coded the slides along 22 theoretical categories, we used step-wise model selection to determine suitable models for variance in neural activity. Preliminary results indicate that the best code-based models at a given instant can account for between 25 and 50 percent of the variance in electrical activity near the EEG electrode for that model when participants observe their own proofs and between 33 and 75 percent during researchers’ proofs.
2017
San Diego, California
Only recently ‘abstraction on objects’ has attracted attention in the literature as a form of abstraction that has the potential to take account of the complexity of students’ knowing and learning processes compatible with their strategy of giving meaning. This paper draws attention to several emerging insights from the evolving framework of structural abstraction in students’ knowing and learning of the limit concept of a sequence. Particular ideas are accentuated that we need to understand from a theoretical point of view since they reveal a new way of understanding knowing and learning advanced mathematical concepts and have significant implications for educational practice.
2017
San Diego, California
Analysis of properties of physical quantities represented by vector fields often involves symmetries and spatial relationships that are best expressed in three dimensional non-Cartesian coordinate systems. Many important quantities, both scalar and vector in nature, are determined by paths, areas, or volume integrals of multivariable functions. The differential quantities in these systems are not trivial for students to understand and implement correctly. As part of an effort to investigate physics students’ understanding of the structure of non-Cartesian coordinate systems and the associated differential elements when using vector calculus in Electricity and Magnetism (E&M), we interviewed four pairs of students in the junior-level E&M course. In one particular task, students were asked to construct differential length elements for an unconventional spherical coordinate system. A symbolic forms analysis (Sherin, 2001) of student reasoning revealed both known and novel forms, and found that student difficulties with vector differential quantities were primarily conceptual rather than symbolic.
2017
San Diego, California
This study investigates Calculus, Transition-to-Proof, and Advanced Calculus students’ meanings for quantifiers in conditional statements involving multiple quantifiers. Three students from each course participated in clinical interviews. Students were presented with the Intermediate Value Theorem (IVT) and three other statements whose sentence structure was similar to the IVT except for reordered quantifiers and their attached variables. The results reveal that Advanced Calculus and Transition-to-Proof students made distinctions between the different statements more often than Calculus students. Several student meanings for quantification were found to be necessary for making distinctions between each of the four statements. We also address student quantifications that emerged for the phrase “Suppose f is a function.”
2017
San Diego, California
Over the past decade research has shown that a Riemann sum based interpretation of the definite integral supports a robust understanding of the underlying structure of the integrand/differential relationship and facilitates students’ ability to make sense of contextual integral models. However, current studies center this understanding on the multiplicative structure݂ f(x) * Delta x which does not account for many practical uses of integration. In many situations, the Delta x is most productively conceived as a component of another quantity which might then be incorporated in any of a variety of quantitative models, such as an inverse square law rather than a simple product. To fill this gap, this study utilized Dewey’s theory of inquiry to identify three interpretations of the definite integral which proved productive for students when modeling definite integrals that extend beyond the traditionally studied product structure.
2017
San Diego, California
We report a qualitative analysis of 14 undergraduate students’ experience in a semester long introduction to proof course. Half were mathematics majors. Our research aims to characterize, conceptually and empirically, students’ transition from a focus on computation to proof in mathematics. Our analysis focused on how students saw the course as different from prior courses, how they described their work in it, and whether being successful in the course required new or different learning activity of them. This approach—targeting students’ overall experience of the course—differs from prior research that has tracked students’ challenges, focused on their work on specific proof problems, and explored how to support and improve their work (e.g., Selden & Selden, 2003). Our work has promise for informing the design of transition to proof courses and how those courses are organized and taught.
2017
San Diego, California
Researchers have demonstrated the importance of covariational reasoning for students’ development of various mathematical ideas. Several researchers have also argued that creating and sustaining multiplicative objects is a necessary mental action to reason covariationally. In this report, we describe task-design principles we have found to be productive for investigating and supporting students’ construction of multiplicative objects and their covariational reasoning. Drawing on our research investigating students’ covariational reasoning, we include data that highlights how these principles have been productive in our research and teaching.
2017
San Diego, California
College mathematics instructors often view the final problem solving steps in their respective disciplines as “just Algebra”, but in reality, a weak foundation in Algebra may be the cause of failure for many college students. The purpose of this paper is to identify common algebraic errors students make in college level mathematics courses that plague their ability to succeed in higher level courses. The identification of these common errors will aid in the creation of a model for intervention.
2017
San Diego, California
As a part of a larger RME-based instructional design project for advanced calculus, this paper reports on two students’ reinventions of formal conceptions of sequence convergence and the completeness property of the real numbers in the context of developing a proof of the Intermediate Value Theorem (IVT). Over the course of ten, hour-long sessions I worked with two students in a clinical setting, as these students collaborated on a sequence of tasks designed to support them in producing a proof of the IVT. Along the way, these students conjectured and developed a proof of the Monotone Convergence Theorem. Through this development I found that student conceptions of completeness were based on the geometric representation of the real numbers as a number line, and that the development of formal conceptions of sequence convergence and completeness were inextricably intertwined.
2017
San Diego, California
This paper reports findings from a study that explored the effect of a secondary mathematics teacher’s level of attention to quantitative reasoning on the quality and coherence of his instruction of angle measure. I analyzed 37 videos of an experienced teacher’s instruction to characterize the extent to which he attended to supporting students in reasoning quantitatively, and to examine the consequences of this attention (or lack thereof) on the quality and coherence of the meanings the teacher’s instruction supported. My analysis revealed that the incoherencies in the teacher’s instruction were occasioned by his inattention to quantitative reasoning. This study therefore demonstrates that when teachers do not possess a disposition to attend to quantities and their relationships, the circumstances are ripe for instruction that emphasizes inconsistent, incoherent, and sometimes incompatible, mathematical meanings.
2017
San Diego, California
Evidence from recent Taylor series studies suggests that well-designed virtual manipulatives can support calculus students in developing an understanding of Taylor series convergence consistent with the formal pointwise convergence definition. In particular, virtual manipulatives depicting convergence along vertical number lines (VNLs) provide graphical representations of quantities necessary for pointwise convergence. We detail one student’s reasoning about Taylor series convergence before and after a VNL was revealed in a Taylor series graph. Prior to the VNL the student had produced very accurate Taylor polynomial graphs based on visually perceptual clues but had omitted notions of pointwise convergence. After a VNL was revealed, the student’s reasoning now included quantities along the vertical as he responded to approximation tasks. We believe that such reasoning can later support the student in developing an understanding of pointwise convergence.
2017
San Diego, California
In this paper, I share results of a case study describing the development of two undergraduate students’ geometric reasoning about the derivative of a complex-valued function with the aid of Geometer’s Sketchpad (GSP). My participants initially had difficulty reasoning about the derivative as a rotation and dilation. Without the aid of GSP, they could describe the rotation and dilation aspect of the derivative for linear complex-valued functions, but were unable to generalize this to non-linear complex-valued functions. Participants’ use of GSP, speech, and gesture assisted with discovering function behavior, generalizing how the derivative describes the rotation and dilation of an image with respect to its pre-image for non-linear complex-valued functions, and recognizing that the derivative is a local property.
2017
San Diego, California
Although studies have shown that students have difficulty with slope and derivative concepts,little is known about connections between these difficulties. In this study, written surveys andclinical interviews were used to examine students’ understanding of both slope and derivative inreal-life contexts. The dominant incorrect reasoning was thinking of slope as the ratio-of-totals
2017
San Diego, California
Previous studies have explored student understanding of vectors in physics, engineering, orlinear algebra settings, but there has been scant research on student understanding of vectors ina multivariable calculus context. In this study, we begin to explore how students think aboutvectors and cross products by analyzing student responses to open-ended questions from anonline, conceptually-oriented multivariable calculus cross product activity. We identify severalthemes consistent with previous research on physics students including confusion between thecross product and its magnitude as well as difficulty identifying or communicating the directionof the cross product vector. This preliminary research begins to develop categories that couldoutline a conceptual model of student understanding of vectors and cross product. The analysisalso informs several recommendations for improving the cross product activity.Key words:
2017
San Diego, California
In this paper we analyze variations in the structure of courses designed for the Precalculusthrough Calculus 2 (P2C2) sequence. We examine the nature of such variations, frequencynationally, and how DFW rates and instructional approach compare to the standard courses.While most identified variations in course structures have on average lower DFW rates whencompared to the national average, a comparison within institutions indicates that thesealternative course structures have higher DFW rates when compared to the standard P2C2sequence offered at the respective institution. In addition, we observed that course variationswhich allow for increased instructional time have greater amounts of active learningtechniques as part of the instructional format. Results from these findings along with theirimplications for the next phase of the Progress through Calculus project are discussed.
2017
San Diego, California
There is a robust body of research demonstrating that when students are asked to justify amathematical assertion, they will frequently generate empirical arguments to do so. They alsosometimes claim a deductive argument does not supply them with certainty that the assertion iscorrect. Mathematics educators frequently attribute this to students having deficient standards ofconviction. In this paper, we illustrate another theoretical account. Students might believe thatthey lack the cognitive capacity to produce a superior argument to an empirical argument or toverify that a deductive argument is correct.
2017
San Diego, California
We present case studies of a student and a non-mathematics professor reading an excerpt from a calculus textbook. We use the ideas of sense-making frames and gaps and the implied reader to compare their reading experiences. In particular, we attempt to distinguish the role of calculus background knowledge from reading expertise in making sense of the text.
2017
San Diego, California
To develop graduate student instructors’ (GSIs) skills and abilities as collegiate mathematics instructors, researchers at two universities implemented a peer-mentorship model where experienced GSIs completed a 15-week professional development (PD) to learn how to mentor novice GSIs in teaching undergraduate mathematics. Using pre-survey, post-survey, and semi- structured reflective interviews, we studied changes in 11 mentor GSIs’ perspectives on teaching and learning practices and what aspects of the mentor PD were deemed valuable by the mentors. Results suggest that this mentor PD, as a peer-mentorship model, helped GSIs deconstruct the dichotic mathematical paradigm of statements being true or false when discussing teaching. Moreover, mentor GSIs valued how the mentor PD helped guide them to facilitate novice GSI post-observation discussions.
2017
San Diego, California
I describe reactions of secondary school mathematics teachers to the following assertion: “According to the established order of operations, division should be performed before multiplication”. I use the notions of local and nonlocal mathematical landscape (Wasserman, 2016) to analyze teachers’ responses to the convention of order of operations in general and the presented assertion in particular.
2017
San Diego, California
Mathematicians commonly distinguish two modes of work in the discipline: Problem solving, and theory building (Gowers, 2000). Mathematics education offers many opportunities to learn problem solving. This paper explores the possibility, and value, of designing instructional activities that provide opportunities to learn mathematics theory-building practices. It begins by providing a definition of these theory-building practices on the basis of which to formulate principles for instructional designs. The paper argues that theory-building practices serve not only the synthesizing role that they play in disciplinary mathematics, but they also have the potential to enrich learners’ reasoning powers and to enhance their problem solving skills. These instructional designs offer a new approach to supporting student work on generalization and abstraction. They have been piloted with preservice and practicing secondary teachers.
2017
San Diego, California
In Where Mathematics Comes From, Lakoff and Núñez (2001) describe how the notions of infinity, continuity, and limit can be constructed through metaphorical extensions of embodied experiences. This paper will critique their historical and psychological analysis, revealing an unresolved tension between a simplified, geometric “approaching” conception and the arithmetization of calculus by Weierstrass. A proposal of how to rectify this conflict through acknowledging how novices can metaphorically tie these concepts together is discussed.
2017
San Diego, California
This theory-based report gives evidence and builds a conceptual framework for a construct called “mathematical knowledge for teaching future teachers” (MKT-FT). Mathematics teacher educators construct MKT-FT as they teach courses for pre-service teachers. Connections to mathematical knowledge for teaching (MKT) are discussed, with an emphasis on the complex relationships between aspects of pedagogical content knowledge in MKT-FT and MKT.
2017
San Diego, California
The terms equity, diversity, inclusion, and social justice have entered the research lexicon. Yet, researchers face significant challenges in gaining a nuanced understanding of the various ideas associated with these words. This theory-focused report presents some recent policy efforts to generate a shared meaning for “social justice” in mathematics education and offers a framework for making sense of (and making sense with) intercultural interactions as an essential component of rigorous research. To anchor discussion, we focus on research on teaching and learning in the courses before calculus (e.g., algebra, pre-calculus, liberal arts math, math for pre-service elementary school teachers, algebra-based statistics).
2017
San Diego, California
A primary function of mathematics education is that students understand the subject matter ofmathematics. That is, students are supported in understanding mathematical concepts andattaining mathematical knowledge. But there is another function of mathematics education, oftenunaddressed in research, which deserves more attention. In addition to learning content, studentsmust be supported in developing informed views about the human processes by whichmathematical knowledge is produced and the unique characteristics of that knowledge. Throughan exploration of humanistic philosophy of mathematics, the purpose of this paper is to identifycharacteristics of the nature of mathematical knowledge that may be important for undergraduatemathematics majors to know and understand. Four characteristics are discussed: mathematicalknowledge is subject to revision; mathematical knowledge is socially validated; proofs are bearersof mathematical knowledge; and informal mathematical work is the foundation of formalknowledge.Key words:
2017
San Diego, California
Overwhelming evidence favors the use of active learning in undergraduate STEM classrooms.Thus, the issue faced by educators is no longer what to do in classrooms, but how to enact whatis known to be effective. This poses a challenge, because faculty teaching is embedded in thecontext of departments, universities, and the broader disciplinary culture. Thus, improvingeducation requires knowledge of how systems work and how to enact systemic change. Whileorganizational change has studied these issues for decades in nonprofit and business settings,the application of this knowledge to higher education is relatively new. Accordingly, thistheoretical paper provides an introduction to the organizational change literature in the contextof higher education and provides an example of its application through Departmental ActionTeams (DATs). By highlighting five principles from organizational change, this paper serves asa reference for change agents wishing to improve undergraduate mathematics education.
2017
San Diego, California
Although many policy documents include equity as part of mathematics education standards and principles, researchers continue to explore means by which equity might be supported. Teaching practices that include active learning have been proposed to address this issue (e.g., CBMS, 2016; NCTM, 2014). In this paper, we theoretically explore the ways in which active learning teaching practices that focus on teaching for inquiry (e.g., Inquiry-Based Learning (IBL) or Inquiry-Oriented Learning (IOL)) support equity in the classroom. Specifically, we claim that some characteristics of inquiry (Student Ownership, Knowledge Building, Peer-Involvement, Doing Mathematics, Student-Instructor Relationship, and Student Success) put forth by Cook, Murphy, and Fukawa-Connelly (2016) may align with the Four Dimensions of Equity (Access, Achievement, Identity, and Power) proposed by Gutiérrez (2009). Therefore, inquiry teaching may be a first step for a focus on equity without compromising the excellence (Gutiérrez, 2002) or material that is often prescribed in undergraduate mathematics courses.
2017
San Diego, California
Although it is frequently a required course, many secondary teachers view real analysis asunnecessary and unrelated to teaching secondary mathematics. In accord with a proposed modelfor improving the teaching of advanced mathematics courses for teachers, we implemented acourse that framed real analysis content by ‘building up from’ and ‘stepping down to’ teachingpractice. In this paper, we describe how this model was implemented in a single module andanalyze secondary mathematics teachers’ engagement in and reflections on the desiredpedagogical aims, which provide evidence that they saw what they learned in the real analysismodule as being useful for informing their pedagogical practice.Key words:
2017
San Diego, California
This preliminary report offers initial results from a study designed to begin identifyingcharacteristics of digital literacy in mathematics. Undergraduate students in a three-coursehonors calculus sequence were provided with tablet computers as part of a digital literacyinitiative and digital tasks were integrated into the courses. Student work was analyzed andcoded for type of ICT tool use and possible components of mathematical digital literacy. Thespecific types of tasks developed for and integrated into the class will be discussed below withspecific illustrative examples highlighted. The aspects of mathematical digital literacyilluminated by student work will be outlined, with some initial conclusions and conjectures aboutthe nature of digital literacy in mathematics.Keywords:
2017
San Diego, California
This work investigates the following research question: How do non-major students understandand use mathematics to solve chemical kinetics problems involving integrated rate laws?Personal constructs, a blend of personal and social constructivism, serves as the theoreticalframework for this study. Semi-structured interviews with 36 general chemistry students, 5upper-level physical chemistry students, and 3 chemical engineering students were conductedusing a think-aloud protocol. Audio and written data were collected using a Livescribe pen. Theaudio data were transcribed, and screenshots of students’ written data were inserted into thetranscripts; these transcripts were refashioned into problem-solving maps. Open coding of theproblem-solving maps reveals initial themes regarding students’ understanding and use ofmathematics when solving chemical kinetics problems. Blended processing was used as amethodological framework to guide the coding process. Through this analysis, distinctive typesof blended processing have emerged.Key words:
2017
San Diego, California
Geometry is the subject where U.S. students are weakest on international assessments, but college geometry is an area of proof that is understudied. Since geometry is secondary students’ only exposure to proof, it is vital our secondary teachers can prove effectively in this content area. The purpose of this case study, drawn from a larger project, was to understand how, if at all, pre-service teachers’ proof schemes became more axiomatic throughout a one-semester inquiry-based college geometry course. Participants in this study, Kayla and Lindsey, were pre-service teachers enrolled in an inquiry based college geometry course. Although Kayla had two prior proof courses and Lindsey had none, both participants were using a perceptual proof scheme at the beginning of the semester. However, by the end of the semester, the chance to revise their proofs and discuss problems with their peers helped both students advance to more axiomatic geometric thinking.
2017
San Diego, California
Traditional training programs that address mathematics graduate teaching assistants’ (MGTAs)teaching practices are offered when they first arrive to campus, when they have little, if any,teaching experience. However, not much research has investigated how MGTAs’ thinking about andfacility with teaching change over the course of their graduate programs and, consequently, howtheir need for training changes over time. The goal of this study is to understand MGTAsdevelopmental stages for teaching and how understanding these stages can inform the creation of amulti-year training program. Eleven MGTAs from a large, doctoral granting institution weresurveyed and interviewed over the course of an academic year. Survey and interview responses wereexamined using a specific model of teacher development. Preliminary analyses, suggestions formulti-year MGTA training programs, and questions for future research are discussed.Keywords
2017
San Diego, California
This report will investigate the mathematical and pedagogical consequences of flipping versus folding in the identification of reflection symmetry. Preliminary results are presented from a teaching experiment aimed at exploring the development of one undergraduate student’s understanding of symmetry. The analysis indicated that throughout the teaching experiment, the student held two distinct versions of reflection symmetry. One version was what most would identify as reflection, while the other was an iterative process based on the participant’s ability to fold the figure. In this report, we share what this student identified as symmetries and how she justified her methods. In addition, we discuss why the definition of isometry necessitates a rigid motion and how the motion of folding is insufficient for identifying symmetries correctly. Lastly, we consider why understanding a rigid motion is advantageous for students who want to consider symmetries in more sophisticated mathematical contexts such as group theory.
2017
San Diego, California
This study examines students’ procedural and conceptual understanding as evidenced by theirwritten responses to two questions designed to assess aspects of their understanding ofeigenvalues and eigenvectors. This analysis draws on data taken from 126 students whoseinstructors taught using a particular inquiry-oriented instructional approach and 129 comparablestudents whose instructors did not use this instructional approach. In this proposal, we offerexamples of student responses that provide insight into their reasoning and summarize broadtrends observed in our quantitative analysis. In general, students in both groups performed betteron the procedural item than on the conceptual item. Additionally, the group of students who weretaught with the inquiry-oriented approach outperformed the group of students who were taughtusing other approaches.Key words:
2017
San Diego, California
In this preliminary proposal, we report on results from a paired-student teaching experimentfocused on college calculus students’ developing notions of reversibility and reciprocity throughcompositions and transformations of linear relations. We anticipate fruitful discussions aboutrelationships between numerical and quantitative reasoning and students’ thinking aboutgraphs.Key words:
2017
San Diego, California
Physics students struggle to make meaning of the negative sign in a variety of mathematical andphysical contexts. This study is part of an ongoing concurrent mixed methods exploration ofstudent understanding of negativity in physics. A set of multiple-choice items, modified from aprior study, was administered to over 500 calculus-based college students from diversebackgrounds. Results suggest that when the positive sign is an explicit part of a quantity,students struggle with positive quantity just as they do with negative quantities, and that thelanguage that instructors use may inadvertently impute unintended meaning about signs.Key words:
2017
San Diego, California
Proof is central to the curriculum for undergraduate mathematics majors. Despite transition-to-proof courses designed to facilitate the transition from computation-based mathematics to proof-based mathematics, students continue to struggle with mathematical proof. In particular,research suggests that proof by contradiction is a difficult proof methods for students toconstruct and comprehend. The purpose of this paper is to discuss preliminary results on studentcomprehension of proof by contradiction within a transition-to-proof course. Grounded in APOSTheory, this paper will illustrate that
2017
San Diego, California
Textbook authors and instructors choose how to define the concept of function for students. Thisstudy examines the impact of definition choice on the mathematics work of graduate students, allof whom were mathematics majors and most of whom are in-service mathematics teachers. Dataare student work on tasks requiring the application of different textbook definitions of functions.By drawing on ideas about action vs. object conceptions of function, it is hypothesized that certainlinguistic features of definitions may affect students’ abilities to use the definition and to build arobust concept image of function.Key words:
2017
San Diego, California
Hypothesis testing is a key concept included in many introductory statistics courses. Due tocommon misunderstandings of both scientists and students, the use of hypothesis testing tointerpret experimental data has received criticism. This paper describes preliminary resultsobtained from a larger study designed to investigate introductory statistics students
2017
San Diego, California
Despite concerted efforts on the part of educational policy makers, women are stillunderrepresented in the STEM fields. Researchers have shown that calculus plays a major rolein this gender disparity since it requires spatial skills to succeed: skills that women tend to lackcompared to men. However, previous studies have shown that spatial ability is malleable andspatial skills can be improved with training. This pilot study employed spatial training in a third-term calculus course and measured the effects of this training on students’ calculus ability,spatial rotation ability, and cognitive learning style. Associations between cognitive learningstyle and task performance were also measured. Preliminary results indicate that spatialtraining does not significantly impact student performance on a calculus skills assessment or atest of mental rotations, but effects on students’ cognitive learning style are present.Key words:
2017
San Diego, California
This paper presents preliminary results of using variation theory to design modeling tasks in order to explore ways of strengthening undergraduate engineering students’ modeling skills. The responses of two undergraduate engineering students enrolled in differential equations to a set of three versions of the same task are reported.
2017
San Diego, California
One of the fundamental concepts in mathematics is that of a function. This concept also appearsto be a difficult concept to grasp for a large percentage of students. In order to assess the overallunderstanding of the concept of a function, we conducted an experiment with math majors at ouruniversity. In an upper division math problem solving course, the students were asked specificquestions about the nature of functions. Students presented their understandings of function ingroups of 2-3, which were recorded and then transcribed. Based on the IBL teachingmethodology and the small group and classroom discussion data collected we have applied asociocultural framework. The innovation we add is applying APOS, an individually orientedtheory, to the collective level. Analyzing the video transcripts, we will discuss the overall trendsin understanding as well as some of the common misconceptions that we have identified.Keywords:
2017
San Diego, California
Despite recent emphases on teaching that mathematics is useful, little is known about teachers’beliefs about the value of the topics they teach. While teachers have been found to believemathematics overall is worthwhile, it may be that this belief varies among subjects. This studyexamines the beliefs and knowledge of students enrolled in teacher preparatory courses byemploying a survey and algebra and geometry tasks related to each subject’s usefulness andconnection to real-world applications. Preliminary results show they value algebra abovegeometry in terms of future use by their students. However, while their confidence was equal intheir abilities to produce real-world applications, they were more successful producing thoserelated to geometry than algebra. Further survey and clinical interview data collection isplanned with additional pre-service and in-service teachers to examine beliefs and knowledgeexpert (in-service) teachers bring to teaching to inform the preparation of secondary schoolteachers.Key words:
2017
San Diego, California
Little is known about the difficulties second semester calculus students have determining seriesconvergence, and why students have such difficulty. This report seeks to add to the existingliterature on series by analyzing second semester calculus student responses to a multiple choiceitem that involves the use of the contrapositive of the nth term test. We frame our discussion interms of what these answer choices might say in terms of student concept images of series andsequences. We also analyze what prerequisite knowledge might help students be more successfulin answering questions about series and sequences typically seen in a second semester calculuscourse.Key words
2017
San Diego, California
Researchers who evaluate efforts to improve STEM undergraduate education have recently begun to explore the importance of instructors’ informal teaching discussion networks. These informal networks allow for the flow of knowledge between instructors that can include information about how to implement research-based instructional practices and creative perspectives that lead to innovative solutions to address localized classroom challenges. In this report, we reanalyze the network data from three pioneering studies in this area to explore the features of mathematics department networks as compared to other STEM department networks at multiple institutions. We plan to discuss implications of these features on the design and implementation of change efforts.
2017
San Diego, California
Raising Calculus to the Surface is a multi-year project designed to introduce important topics frommultivariable calculus through the use of physical manipulatives. This report focuses on datacollected through a series of task-based interviews with multivariable calculus students enrolledin a course featuring these manipulatives. To explain the students’ activity, a two-dimensionalframework was designed based upon characterizations of their interaction with the instrumentsand the generality of their mathematical activity. The report concludes by discussing thecontributions to the field and possible future uses of the framework.Key words:
2017
San Diego, California
Self-inquiry is the process of posing questions to oneself while solving a problem. The authors’previous work has explored the self-inquiry of undergraduate mathematics majors and amathematics professor. Student self-inquiry was explored via structured interviews requiring thesolution of both mathematical and non-mathematical problems. The professor’s self-inquiry wasexplored 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 developedand extended after coding the professor’s self-inquiry. Previous results will again be highlightedhere but will be followed by a discussion of self-inquiry in the context of an introduction tomathematical proof course. Data from the introduction to proof course is being collected andwill be analyzed using the already developed coding scheme. This analysis will be compared andcontrasted to previous self-inquiry results and we will present questions about possible futuredirections for exploring self-inquiry.
2017
San Diego, California
Recent research illustrates the importance of studying students’ nuanced mathematicalargumentation, as well as students’ tendency to invoke attributes of real numbers that no longerapply to situations in complex analysis. This preliminary report explicates a study exploringundergraduate student pairs’ reasoning about integration of complex functions. I amparticularly interested in students’ attention to the idiosyncratic hypotheses of powerfulintegration theorems as they evaluate integrals. Here reasoning is treated as contributing tocollective argumentation within one or more of Tall’s (2013) three worlds of mathematics. Datawere collected via task-based, semistructured interviews with pairs of undergraduates to elicitsuch reasoning, and classroom observations of the six class sessions devoted to integration priorto the interviews. All interviews have been transcribed and current analysis consists ofconducting a Toulmin (2003) analysis, augmented by a three-world classification. Potentialimplications of this work and connections to the associated literature are also discussed.
2017
San Diego, California
DNR-based professional development (DBPD) is a long-running program spanning seven yearswith multiple cohorts of in-service secondary mathematics teacher participants. This reportinvestigates teacher change among five key variables: facilitating public debate, using holisticproblems, attending to students’ intellectual need, attending to meaning of quantities and use ofstudents’ contributions. Is there evidence that DBPD contributed to higher implementation amongparticipants over time? What factors afford/constrain DNR implementation over time? Classroomobservation data indicate the largest impact was found in teachers’ attention to meaning ofquantities and students’ intellectual necessity while interview data provide insights to what affordsand constrains DNR implementation.Key words:
2017
San Diego, California
This paper reports on critical aspects of three engineering students’ discourse in group work using a digital tool called Sim2Bil while solving mathematical tasks. Applying a commognitive perspective, where mathematical discourse is characterized by words used, visual mediators applied, narratives developed and routines established, we investigate how these characteristics are influenced by the technological environment. It is found that all of the aspects of the students’ discourse are influenced by Sim2Bil. For instance, a “trial and error” routine directly connected to the use of the tool is present in the students’ discourse.
2017
San Diego, California
Historically Black Colleges and Universities (HBCUs) have a longstanding legacy of supportingAfrican American students in mathematics. The undergraduate mathematics faculty membersplay a unique role in supporting and developing astute mathematics students, especially AfricanAmerican male students. This preliminary research report highlights the experiences of a cohortof 16 African American male mathematics majors at an all-male, private HBCU by investigatingthe role of the mathematics faculty members. Using qualitative research methods grounded incritical race theory, preliminary data show these African American male mathematics majorsbenefited (mathematically and racially) by their supportive mathematics faculty members.
2017
San Diego, California
Drawing from task-based interviews, classroom observation, and participants’ homework, thepresent study examines ten middle grades preservice teachers’ understanding of the role offractions as operators, with an eye toward exploring how fractional reasoning is constructed.The results point to the construction of the reversible distributive partitioning scheme as arequisite for understanding fractions as operators. Further discussion will suggest that schoolcurricula and teacher education programs may need to be adjusted to reflect more currentunderstanding of both early childhood cognitive development and future teachers’ fractionalknowledge.Key words:
2017
San Diego, California
In this study, an advanced undergraduate geometry class taught in an inquiry-based learningsetting was observed for social and socio-mathematical norms. Three pairs of students engagedin three task-based, semi-structured interviews: paired, individually, then paired again, solvingthe Seven Bridges of Königsberg and related tasks. A fourth stimulated-recall interview wasperformed using episodes from the last paired interview. Classroom observations and interviewdiscourses were open coded for themes, structure, and function to analyze the norms developedwithin the classroom and by each pair as shaped by their social interactions. Tentative findingsinclude: 1) norms of consensus, autonomy, and argumentation produced within the classroom, 2)varying metaphors across interview contexts, and 3) reliance on empirical strategies rather thanstructural reasoning. In this preliminary report, evidence from collected data is shared and abrief discussion how these results could help inform IBL teaching methods is included.Keywords:
2017
San Diego, California
Improvement of mathematics courses in the first two years of college has recently become apriority in the United States. This is evidenced by multiple calls to enhance undergraduateeducation in the mathematical sciences and by funding allocated to related research andinstructional improvement projects. As stakeholders make decisions to invest in the improvementof these courses, it is critical that these decisions be informed by reliable information regardinghow these courses are currently being taught. The work described here is an effort to lay thisgroundwork by painting a comprehensive portrait of instruction in precalculus and singlevariable calculus (P2C2) in the United States. In this report we address two research questions;1) What instructional formats are currently in place in the P2C2 sequence? and 2) How commonare these instructional formats nationally?Key words:
2017
San Diego, California
This is a preliminary report on a study to investigate the inclination of Calculus III students to use visual reasoning in problem solving situations. One of our research hypotheses was that there is a correlation between students’ inclination towards visual representations when taking notes and their use of visual representations in problem solving situations. Surprisingly, preliminary analysis of the results suggests that there may not be a correlation, although work is ongoing.
2017
San Diego, California
In this paper, we present a comparative case study of two students with different epistemologicalframes watching the same real analysis lectures. We show the general point that students withdifferent epistemological frames can interpret the same lecture in radically different ways. Wealso identify epistemological frames that are useful or counterproductive for understanding alecture on how the rational numbers are constructed from the integers. These results illustratehow different students interpretations of a lecture are not inherently tied to the lecture, butrather depend on the student and that student’s perspective on mathematics. Thus, improvingstudent learning may depend on more than improving the quality of the lectures, but alsochanging student’s beliefs and orientations about mathematics and mathematics learning.Key words:
2017
San Diego, California
The research aims at introducing modelling tasks in order to engage students more actively into learning mathematics through tasks that are biologically ‘colored’. My focus is on the individual progression (if there is any) of students’ mathematical competencies during a sequence of modelling sessions that will be part of a regular course of their first year calculus. My ultimate goal is to construct a dynamic competence profile for every student that will participate in the project. Taking the above into consideration, my research suggests a number of interventions in a standard freshmen mathematics course for biology students, interventions that offer a fruitful didactical environment where students can sharpen their mathematical competencies.
2017
San Diego, California
In a recent combinatorics-focused teaching experiment with two undergraduate students, thestudents developed a robust understanding of a three-stage counting process that provided asolution for problems involving combinations. So strong was the students’ three-stage process,they did not seem to naturally conceive of the singular process of “choosing,” which is animportant aspect of understanding combinations. In this preliminary report, I question whetheror not the students engaged in reification, which Sfard and Linchevski describe as “our mind’seye’s ability to envision the result of processes as permanent entities in their own right”(1994, p.194). I raise questions about what aspects of the student work might have fostered or hinderedtheir ability to reify choosing, as well as what might be taken as evidence that reification hasoccurred in the context of combinatorics.Key words:
2017
San Diego, California
Low success rates in the pre-college level, or developmental, curriculum at many community colleges has resulted in the creation of classes that use problem solving and group work to help students become more mathematically empowered. This preliminary report describes one such class at a Midwestern community college and then outlines the results from a pre- and post- survey of students taking the class, focusing on whether students’ attitudes towards mathematics changed while enrolled in the class. Further analysis will examine how students evaluated the class and ranked the class structures. Generally, males, younger students, and Black students were less likely to complete the course. Students who came close to completing the class had an overall positive shift in their attitudes towards mathematics.
2017
San Diego, California
Retaining mathematically talented underrepresented students in mathematics programs requires understanding the challenges the face during their post-secondary mathematics education. Using Swail’s framework (2003), this study investigates the self-identified challenges undergraduate and graduate mathematics students face and the coping mechanisms that helped them navigate and overcome those challenges. The vast majority of the challenges both groups of students encountered were cognitive in nature, suggesting that programs wishing to retain students should focus on providing social and institutional supports to provide balance.
2017
San Diego, California
The Second-Derivative Test and optimization can naturally evoke gestures from an instructor while he or she is teaching. We wanted to establish how student learning might be affected by an instructor’s use of gesture. Students viewed either a gesture-rich or gesture-free video of an instructor solving an optimization problem, and were interviewed a week later to assess both their understandings of optimization and how they used gesture to support their explanations. Very few gestures were used when the students explained how they solved the optimization problem. However, when describing the second derivative test separate from optimization, students used a number of gestures. We conclude that further study should be undertaken, but such study should be focused on the Second-Derivative Test without the context of optimization problems.
2017
San Diego, California
The mathematical symbol “dx” is a symbol for which there can exist different views about its characteristics, purposes, and roles. We wished to see how experts viewed the dx in a variety of settings. We chose four mathematical contexts and interviewed four mathematics professors in order to understand their various concept images of the dx. While there was little agreement among the experts’ responses, most of them did have a strong concept image that remained consistent throughout their interviews, despite our attempts to create cognitive conflict between the different mathematical contexts. We conclude that the existence of a range in the experts’ opinions is noteworthy, and that further study should be conducted in order to more fully explore this range and any implications for instruction that may result from it.
2017
San Diego, California
In this paper, I describe a teaching experiment conducted with a pair of undergraduate studentsat a two-year community college. My primary goal was to explore a realistic starting point forthe guided reinvention of the concept of limit at infinity for students who had not yet studiedlimits. The teaching experiment included 5 weekly hour-long sessions in which the two studentswere presented with tasks that involved describing the behavior of certain real-worldphenomena. The initial analysis revealed that these students showed ways of thinking thatanticipate the formal concept of limit at infinity. Further analysis will be used to develop anappropriate instructional sequence with a realistic starting point to be used in future teachingexperiments in which students will be engaged in the guided reinvention of a formal definition oflimit at infinity.Key words
2017
San Diego, California
Binary Operations are essential to many undergraduate mathematics courses. However, little isknown about student conceptions around binary operation. This report presents preliminaryresults from nine student surveys about the topic. The question set was developed in response toGroup Concept Inventory (GCI) results. We look at three activities closely related to binaryoperation: identifying when an instantiation is a binary operation, identifying when twoinstantiation are the same binary operation, and generating an original binary operationinstantiation. We use the lens of variation theory to make sense of student responses. We foundthat students’ concept image of binary operation may be missing key attributes (such asrequiring two inputs) and contain unnecessary attributes (requiring a general rule.)Key words:
2017
San Diego, California
Across the nation, there is increased national interest in improving the way mathematicsdepartments prepare their GTAs. In particular, this research focuses on how the mentor GTAs inthe graduate teaching assistant program under consideration share effective teaching practicesand how this effects changes in the teaching practice of GTAs. I report preliminary results onhow the focus of particular teaching practices of mentor GTAs (known as lead TAs) change overthe period of one term through their participation in professional development. With anunderstanding of the differences and the similarities between the focuses of the lead TAs, ananalysis of the differences between the Calculus I and II GTAs will become more apparent. Theresearch presented here represents the start of an increased understanding of how GTAs formtheir own teaching practices.
2017
San Diego, California
Pre-service and in-service high school teachers often do not leverage their experience withabstract algebra when interpreting the notation of inverse functions. For this study, we havedesigned a professional development activity in which teachers can explore inverses in differentsets with different binary operations to elicit pseudo-empirical abstraction of the relationship
2017
San Diego, California
The noun independence and adjective independent are applied in multiple mathematicalcontexts. In probability, independent events do not affect each other, but in algebra andregression, an independent variable has a non-symmetric effect on a dependent variable. Furthercomplicating matters, independence in everyday language represents something in between.Prior research has shown that students and professors struggle to apply concepts ofindependence. As part of an investigation into curriculum about independence, textbookdefinitions about independence were examined. Across nine books, a mix of algebra andstatistics texts, substantial variations existed in definitions of independent events andindependent variables. Variations included the register of representation, verbal againstalgebraic, and the strength of the dependent effect. Little written guidance was provided to helplearners navigate across the multiple formations.
2017
San Diego, California
This study explores how practicing teachers make connections between secondary and tertiary mathematics. Using three frameworks for teacher knowledge of mathematics, coupled with key developmental understandings (KDUs) (Simon, 2006) as related to teacher knowledge (Murray & Wasserman, 2016), we observe how a professional development workshop focused abstract algebra impacts teachers’ understanding and teaching of secondary mathematics.
2017
San Diego, California
Research shows that low-achieving students are less able to accurately assess their ownweaknesses. As a result, many might fail to see the need to explore the subject matter moredeeply, in order to improve their conceptual understanding and procedural fluency. Thisstudy investigates undergraduate mathematics students’ self-assessment behaviors. Studentsfrom a broad range of courses at three universities were asked to predict their expectedgrades on assignments, and these predictions were compared with the grades assessed bytheir instructors. They were also asked to justify their self-assessments if they did not givethemselves full points. Preliminary results showed that students overall overestimate theirgrades. There was a significant difference between expected and actual grades. As test scoresincreased, the difference increased from negative to positive. Students in the B-range(between 80-89%) were the most accurate predictors.Key words:
2017
San Diego, California
We investigated a peer role model intervention designed to alleviate underrepresentation ofwomen in STEM. Half of the Calculus break-out sections at a large university were visited by apeer role model and half served as controls. The female peer role models were expected toincrease the sense of belonging and mathematical self-efficacy of women highly identified withmathematics. Our results show that peer role models have the intended effect on women highlyidentified with mathematics, but also have a positive effect on men with low mathematicalidentification.Key words:
2017
San Diego, California
In a previous study we sought to understand the classroom activities that provided students theopportunity to engage in the five roles of proof described by Michael de Villiers (1990). Inconducting the analysis for that
2017
San Diego, California
The advancement of technology has significantly changed the practices of numerous professions, including teaching. When a school first adopts a new technology, established classroom practices are perturbed. These perturbations can have both positive and negative effects on teachers’ abilities to teach mathematical concepts with the new technology. Therefore, before new technology can be introduced into mathematics classrooms, we need to better understand how technology affects instruction. Using interviews and classroom observations, I explored perturbations in mathematical classroom practice as an instructor implemented novel didactic objects. In particular, the instructor was using didactic objects designed to lay the foundation for developing a conceptual understanding of rational functions through the coordination of relative magnitudes of the numerator and denominator. The results are organized according to a framework that captures leader actions, communication, expectations of technology, roles, timing, student engagement, and mathematical conceptions.
2017
San Diego, California
In this study we focus on the use of graphical representations to find similarities and differencesregarding how graphs are used in mathematics textbooks and how they are used in STEtextbooks and journals. After highlighting the need for our study and summarizing the results ofrelated studies, we present our methods. We then present key preliminary findings comparinghow a selected pre-calculus textbook and certain textbooks and journals in various STE fieldsuse graphical representations. We conclude with preliminary implications and questions.Keywords:
2017
San Diego, California
In the study presented in this paper, the authors aim to construct a model for the processes by which students in a multivariable calculus class conceptualize solid regions in three dimensions. We designed and recorded student work from several tasks in which students must decode a de- scription of a solid figure and answer questions assessing the strength of their conception of the figure. Presented here are findings from the analysis interviews and group work on one of these tasks in which students are asked to build a clay model of the solid region described by a set of inequalities in three variables.
2017
San Diego, California
Much work has been done in recent years to study students’ formulations of formal limiting processes. One of the most common goals is to foster a productive understanding of the relationship between the error bound epsilon and the domain of the convergence; what is called a range-first perspective. My study examined an advanced calculus student’s understanding of the relationships involved in convergence of functions, and how his prior experience with limits influenced his concept image. I unpack his cognitive organization of the dependence relationships between epsilon, N , and x in functional convergence. This case study demonstrates the effects of a persistent understanding that epsilon depend on N in the convergence of sequences.
2017
San Diego, California
Tutoring centers are common in universities in the United States, but the effects of tutoring onstudent success are often not examined statistically. This study utilizes multiple regressionanalysis to model the effect of tutoring attendance on final course grades in Calculus I. Ourmodel predicted that every three visits to the tutoring center would increas
2017
San Diego, California
Functions of the form ! ! = (! ! )!(!), including constant functions, power functions, and exponential functions, are fundamental examples of functions that differential calculus students should be able to differentiate. Yet students often struggle to distinguish between these forms. Drawing on APOS (Action-Process-Object-Schema) theory as well as Piaget and Garcia’s triad of schema development, this paper offers a genetic decomposition of the schemas students build for determining the derivative of a function to a function power. In particular, we analyze how students determine which differentiation rules to use with different function structures of a function to a function power and how students construct a conception of logarithmic differentiation. An initial genetic decomposition informed by existing literature was refined using the results of a series of three clinical interviews with each of two calculus students. Findings include the necessity of a strong background in functions, logarithms, and other differentiation rules.
2017
San Diego, California
Some mathematics education publications highlight the importance of fostering students’ mathematical creativity in the undergraduate classroom. However, not many describe explicit instructional methodologies to accomplish this task. The authors attempted to address this gap using a formative assessment tool named the Creativity-in-Progress Rubric (CPR) on Proving. This tool was developed to encourage students to engage in practices that research studies, mathematicians, and students themselves suggest may promote creativity in processes of proving. Three instructors in different institutions used a variety of tasks, assignments, and in- class discussions in their proof-based courses centered around the CPR on Proving to explicitly discuss and foster mathematical creativity. These instructors’ actions are explored using Levenson’s four teacher roles of fostering mathematical creativity. In this report, preliminary results indicate that each of the three instructors assumed at least three of the four roles.
2017
San Diego, California
This case study continues the story of the development of Alice’s proof-writing skills into the second semester. We analyzed the videotapes of her one-on-one sessions working through our inquiry-based transition-to-proof course notes. Our theoretical perspective informed our work and includes the view that proof construction is a sequence of mental, as well as physical, actions. It also includes the use of proof frameworks as a means of initiating a written proof. Previously, we documented Alice’s early reluctance to use proof frameworks, followed by her subsequent seeming acceptance of, and proficiency with, them by the end of the first semester (Benkhalti, Selden, & Selden, 2016). However, upon first encountering semigroups, with which she had no prior experience, during the second semester, her proof writing deteriorated, as she coped with understanding the new concepts. But later, she began using proof frameworks again and seemed to regain a sense of self-efficacy.
2017
San Diego, California
This paper reports the results of four groups of three pre-service teachers working on a task that had them investigate the fairness of dice. The teachers used an online collaborative environment to sample from six different dice and the environment provided them with various representations, which they used to support their arguments. All four groups preferred the frequency table over bar and pie charts representations. After working on the task, they evaluated the work of students on the same problem. They viewed students’ work that used other representations as not convincing regardless of the correctness of their solution and showed preference to the students only using one representation. Implications for pre-service teacher training in statistics and how to promote the use of multiple representations are discussed at the end.
2017
San Diego, California
We engaged five research mathematicians in describing their images of differentiation and integration for functions of complex variables. Analyzing the data in terms of Tall’s three worlds, we explore the connections between the physical embodiments of the mathematicians’ reasoning, their descriptions for students, and their formalizations of these interpretations. For differentiation, the mathematicians relied heavily on direct application of concepts and analogies from differentiation of real-valued functions and employed rotation and stretching as a local linear description of the action of the function, corresponding to repeated mental imagery and physical gestures. For integrals, the mathematicians employed reasoning about line real- valued integrals, but acknowledged that they struggled to conceptually interpret what was being accumulated in the complex case. Instead, they all developed more personal meanings through a process of reconciling various aspects across their own conceptual-embodiment, operational- symbolic, and axiomatic-formal reasoning.
2017
San Diego, California
Findings from a recent national survey indicate that two thirds of graduate-degree-grantingmathematics departments provide some form of teaching-related professional development totheir graduate students. Despite the prevalence of such programs, little is known about howdepartments evaluate the quality of the graduate students’ instruction or the efficacy of theirprofessional development. We present a mixed-method analysis of data to shed light on both ofthese topics. We found that graduate students and their professional development are most oftenevaluated based on student evaluations. Other research indicates the ineffectiveness of studentevaluations as measures of teaching, and so this finding indicates a need for research-guidedevaluation tools for graduate student professional development.Key words:
2017
San Diego, California
As colleagues in a Mathematics/Computer Science department, we found that many of ourundergraduates were not able to participate successfully in the full range of STEM courseofferings. In response to this need, we developed a strategy for explicit instruction inmathematical generalization. Our instructional design is grounded in a theory of mathematicallearning that uses computer programming to induce students to build the mental frameworksneeded for understanding a math concept. The design includes writing mini programs to explorea mathematical concept, finding general expressions in the code, making conjectures about therelationships among general expressions, and writing logical arguments for the conjectures. Weshare results from a study of 18 undergraduate math/secondary education majors. Our resultsindicate most pre-service teachers showed improvement in their level of abstraction over theconcept of direct variation.Key words:
2017
San Diego, California
In this study, we examine the use of assigning articles published in NCTM’s practitioner journalsas readings in mathematics content courses for prospective elementary teachers (PTs). Inparticular, we study the articles’ roles in motivating PTs to engage in their content courses. As aconceptual foundation, we characterize NCTM articles as having potential to (1) increase PTs’“buy-in” of pedagogical approaches used in content courses, (2) challenge PTs’ unproductivebeliefs about mathematics, and (3) address mathematics content via children’s thinking. We planto analyze an existing dataset of PTs’ online typed responses to assigned NCTM articles toidentify whether and how their responses reflect increased motivation to engage in their contentcourses. We anticipate that our results will lead to an increased understanding of PTs’ actualexperiences related to the assigned article readings.Key words
2017
San Diego, California
The CCSSM emphasize statistical concepts for grades 6-12. A key factor in thinking statisticallyis to reason about variation and variability. This paper will present the analysis of surveyquestions and tasks given to in-service middle school teachers. The paper will attempt to answer
2017
San Diego, California
Active learning practices highly depend on students’ preparation for class in advance. However, reading Calculus can be a challenging task to students. We address this concern by assigning targeted pre-class readings and reading quizzes in two Calculus II classes. To study the effectiveness of these, we also provided them as post-class readings in two other classes. We report on our implementation and we discuss students’ feedback about the readings and quizzes.
2017
San Diego, California
Graduate students teach many first year undergraduate mathematics courses, such as CollegeAlgebra and Calculus. In this report, we focus on the opportunities to learn to teach thatgraduate student teaching assistants (GTAs) construct from reflecting on their teachingexperiences. Research in professional development suggests that although reflection isabsolutely essential to improving one’s teaching, teachers have the greatest opportunity to learnfrom their teaching when they can mobilize their interpretations of teaching to inform specificand nuanced future actions. Yet, there are few studies addressing the ways in which GTAsdevelop opportunities to learn from reflection. In this case study, we examine how two graduatestudents developed the ability to link observations of student work to hypotheses about studentthinking and then connect these hypotheses about student thinking to future teaching actions.These reflections were generated as a part of a professional development program for GTAs.Key words:
2017
San Diego, California
Despite overwhelming evidence of the effectiveness of student engagement in instruction, practicing mathematics instructors often use instructor-centric practices even if they value student engagement. Answering a call by Henderson and Dancy (2007) to study the implementations of researched-based curriculum in the classroom, this paper looks at the change in practices and values of instructors utilizing active-engagement activities in multivariable calculus classes. This curriculum incorporates context and multiple representations, and we look for evidence that addresses whether these features facilitate instructor use of student ideas in instruction.
2017
San Diego, California
Eigentheory is a conceptually complex idea whose application is widespread in mathematics andbeyond. Herein we describe the development and use of an extended multiple choice assessmentthat gives us further insight into the ways students think about and understand eigenvectors,eigenvalues, and their related concepts.Key words:
2017
San Diego, California
In this report we shared our preliminary analysis of one student’s meta-representationalcompetence as he engages in solving a quantum mechanics problem involving linear algebraconcepts, namely basis, eigenvectors, and eigenvalues. We provide detail on student A25, whoserves as a paradigmatic example of a student’s power and flexibility in thinking in and usingdifferent notation systems. This preliminary work lends credence to and inspires our conjecturethat strong meta-representational competence (MRC) is necessary not only to be fluent andproficient in the mathematics involved in solving quantum mechanics problems but also todevelop a robust understanding of the quantum mechanics content.
2017
San Diego, California
In this paper we describe a study involving twelve preservice elementary teachers who were attending a community college. The design and implementation of this study were guided by the research question: In what ways do students reason through a sequence of tasks which progressively become more abstract, and which challenge primitive intuitions regarding partitive division? We highlight students’ ways of thinking involved with division that are not easily generalizable, that favor numerical procedures over quantitative reasoning, and which are obstacles to the development of more robust meanings for division.
2017
San Diego, California
We examine the current progress of implementing both corequisite remediation and mathpathways in the state of Oklahoma. In this paper, we discuss the details of these effort and theunderlying needs while providing a national perspective about the reforms. We presentpreliminary data from pilot sections of a corequisite College Algebra course and a new mathpathway for degrees that require significant quantitative literacy but do not require engineeringcalculus. We also present statewide data on student course-taking patterns, degree requirements,and existing institutional efforts that will inform state-level decisions.Key words:
2017
San Diego, California
The long-term aim of this study is to develop a conceptual framework outlining the conceptsnecessary for college students to be able to successfully complete fundamental tasks ofelementary algebra. This paper is a preliminary report of one part of this research, whichfocuses on instructor perceptions of what concepts are fundamental to successful completion ofelementary algebra tasks. The framework presented here is the result of an action researchproject conducted by five college instructors in the U.S. who teach elementary algebra.Keywords:
2017
San Diego, California
Abstract: Students who have persisted in mathematics coursework long enough to be presentin calculus or who enter mathematics at the level of calculus would be expected have morerobust notions concerning their career choices than those who enter developmentalmathematics. In the current work, we give a preliminary comparison of data generated by acareer decision making survey administered to students in a developmental mathematicscourse and to students in a first semester calculus course at a large research university duringthe fall 2015 semester. We consider some initial results for students who switch majors after asemester of mathematics coursework.Keywords:
2017
San Diego, California
The work reported in this paper is part of a study aimed at characterizing the processes andidentifying the ways in which different kinds of expertise (mathematics vs. mathematicseducation) unfolded in the planning and teaching of an undergraduate course on MathematicalProof and Proving (MPP), which was co-taught by a professor of mathematics and a professorof mathematics education. More specifically, the study aimed at unpacking the affordances anddrawbacks of this collaboration. The collected data includes all 13 videotaped lessons in the2012 semester, the second time the course was taught. The content of the course consisted oftopics that were familiar/accessible to the students, e.g., high school level algebra, geometry,and basic number theory. In this paper we focus on how the views held by each instructorregarding what constitutes an acceptable proof and how it should be presented, are reflectedin his/her teaching.Key words:
2017
San Diego, California
This evaluation study compared the mathematical thinking of undergraduate students (as theyresponded to class work and interview prompts) who participated in an inquiry-based linearalgebra course to a comparison group of students who participated in a traditional course.
2017
San Diego, California
We investigated how a teacher’s meaning of constant rate of change influenced his teachingpractices. Findings revealed that a teacher with a strong mathematical meaning of constant rateof change was able to provide conceptually coherent explanations and pose questions that arebased in his understanding of constant rate of change and his models of students’ thinking.
2017
San Diego, California
This study focuses on the concept of including traditional math classroom experiences in a mathemporium course. The aim of the study is to gain an insight into the opinions of students aboutwhich emporium structure they prefer as well as which they believe they can be more successfulin. Also, this study will analyze emporium students’ academic success in both scenarios. Toaccomplish these goals, two sections of Algebra II in the math emporium were offered the optionto attend short instructional opportunities led by the instructor.Key Words
2017
San Diego, California
This poster reports on an action research project set in a developmental mathematics classroomin a community college. Students at the beginning of the semester expressed mathematics anxietyand did not believe they could succeed in the course. To help support students’ learning, theinstructor and students co-created a list of ten statements that became the prevailing philosophyin the class. These statements helped students alter their view of their own mathematics learning.Key words:
2017
San Diego, California
This study explored efforts to design and empirically test measure teachers’ knowledge of the nature of mathematical modeling. The author begins by reviewing the literature on teachers’ content knowledge and mathematical modeling, noting the effect of content knowledge on teachers’ professional competence. Next, the author discusses the items used on the questionnaire to represent knowledge of the nature of mathematical modeling. Results from reliability, factor analysis, and scaling work with the items showed teachers’ knowledge of the nature of mathematical modeling was unidimensional. The construct indicated by factor analysis formed psychometrically acceptable scale for measuring teachers’ knowledge of the nature of mathematical modeling.
2017
San Diego, California
In this study, I designed and implemented an instructional sequence of exploratory activities using a Dynamic Geometry Environment (DGE) in an axiomatic geometry course. The tasks in the sequence aimed at providing students with opportunities to encounter cognitive conflicts between their prior knowledge on Euclidean geometry and new observations on non-Euclidean geometry. However, some did not appear, some did appear and students recognized them, but could not resolve or just passed by. The conflict between what I intended in designing tasks and what I found in student responses seems to result from several aspects of design and implementation of the tasks.
2017
San Diego, California
One of the reasons for the exodus in STEM majors is students’ experiences in their first undergraduate mathematics course, usually introductory calculus. However, students with a growth mindset are more likely to persist past these initial courses. Although there is evidence that curricula like CLEAR calculus promoted gains in students’ growth mindset, it is unclear how this curriculum compares to traditionally. The purpose of this quasiexperimental study was to investigate to what extent students enroll in CLEAR calculus become more growth mindset orientated than those that are enrolled in traditionally taught courses. The Patterns of Adaptive Learning Scale was used to measure the mindset of students in pre-calculus, calculus I, and calculus II. The analysis of the pilot data indicated CLEAR calculus students experience a small positive shift towards a growth mindset, while students in traditionally taught courses have a significantly more fixed mindset by the end of the semester
2017
San Diego, California
Symmetry has been found to be a rich and natural context for developing group theory (Larsen,2009), yet the existing literature offers little insight on the complex cognitive processes in thisdomain. This poster will describe an attempt to use a pre-existing cognitive model of a student’sunderstanding of symmetry, to help analyze the data from a recent teaching experiment aimed atexploring the development of one undergraduate student’s understanding of symmetry. Weshare the ways in which the existing model accurately describes the student’s cognitiveprocesses associated with symmetry and also the places in which the model fell short incapturing the complexity of the student’s thinking.Keywords:
2017
San Diego, California
Research has shown that students have difficulties attending to the underlying product andsummation structure of the integral when solving application problems. This study examinesstudent conceptions of the product layer when solving volume problems. Participants weresecond-semester calculus students enrolled in a large, public university. Task-based interviewsconsisted of students working through and discussing volume problems. Preliminary resultsshow that a majority of students’ volume integral setups are highly formulaic and linked tomemorized patterns and methods seen in class, as opposed to having a true understanding of theunderlying structure of the problem. We plan to conduct more interviews of this type withadditional volume problems and investigate other aspects such as visualization and gesture.Key words:
2017
San Diego, California
In an effort to broaden knowledge within the United States, the National Council of Teachers of Mathematics, with support from the National Science Foundation, funded multiple scholars’ participation in the 13th International Congress of Mathematics Education. Working in NCTM theme groups these scholars met, discussed, and provided reports to various American educational organizations, so as to bring back findings related to a variety of ICME Topic Study Groups. The purpose of this poster is share findings from the “Mathematics Education as a Research Field” NCTM theme group.
2017
San Diego, California
The poster will have three parts. The first is a literature review of reading comprehensionresearch. Connections will be drawn between different research studies that apply to collegemathematics instruction. In addition, common themes from research will be used to motivate theneed for reading comprehension instruction in college mathematics. The second part will giveresults of a new research study involving interviews about reading comprehension strategies andinstruction with interdisciplinary faculty from several institutions. The third part will describeproposed methodology of future research on reading comprehension in college mathematicsinstruction. It is hoped that during the poster session, participants will have a chance tobrainstorm about possible future directions and methodologies for research in this area.Keywords
2017
San Diego, California
Many liberal arts or humanities students who are required to take quantitative reasoning incollege have mathematics anxiety. One cause is a lack of symbolic skills for reasoning. Learningstyle theories suggest that different people learn in different ways. This study constructs adiagrammatic reasoning model for the concept of amortization to help students learnquantitative reasoning. The model also connects the basic concepts to the proof of the mortgagepayment formula.Key words
2017
San Diego, California
We cannot imagine teaching statistics today without using some form of technology. Teaching statistics courses in the past was very challenging due to time consumption in calculation. The computation is now done by computer and other software packages but the challenge now is that understanding the results and staying mistrustful to the results. In this discussion, we will discuss how we have been teaching introductory statistics courses with or without computer and provide some typical examples in excel spreadsheet. On the other hand, due to the recent development of the power of computing, we present a dynamic documentation of computational outputs from a statistical programming language using R markdown (included in the package “knitr”) which is a simple formatting syntax for authoring HTML, PDF, and MS Word documents. Hence the main goal of this presentation is to give an outline of the method used in the past, its challenges as mentioned by Christine Duller (2008) and a demonstration of an R package which recently brought a great attention to the teachers of statistics and researchers. The usefulness of the package will be presented using some data analysis and graphs using R programming language. Since R Markdown supports dozens of static and dynamic output formats including HTML, Pdf, MS Word, Beamer, HTML slides, Tufte-style handouts, books, dashboards, shiny applications, scientific articles, websites, and more, it is more popular to the researcher, teacher of statistics and collaborators.
2017
San Diego, California
Focusing on the unique factorization domain (UFD) in college mathematics and polynomial factorization in school mathematics, this study examined how teachers’ factorization concepts occur in the teaching context. We conducted semi-structured interviews with eight novice teachers. The result of this study can serve as a resource for teacher educators when teaching UFD in abstract algebra in the future.
2017
San Diego, California
As part of a larger effort to develop a research-based math methods curriculum forundergraduate physics students, results from a case study probing student thinking on planeand spherical polar coordinates are presented. Using a resources framework, a think-aloudprotocol was used to elicit student thinking regarding non-Cartesian coordinates. Findings areconsistent with previously published literature regarding student thinking on coordinatesystems. Mark, a senior physics major, despite initially clearly identifying and defining theradial and polar unit vectors on a diagnostic 2-dimensional problem, made inconsistentassertions when asked to apply those definitions in three dimensions using sphericalcoordinates. Additionally, we will address content issues concerning the definition ofdisplacement and position vectors in Cartesian and Non-Cartesian coordinate systems.Key words:
2017
San Diego, California
Student attitudes about and perceptions of mathematics influence their success and learning, andhave been of interest for many years in mathematics education. The Mathematics Attitudes andPerceptions Survey is a short, validated Likert-scale instrument that measures confidence, interest,relation of mathematics to the real world, persistence in problem solving, growth mindset, use ofsense-making behaviours, and the extent of other novice attitudes towards mathematics. In thisposter, we share the complete instrument and its categories, a brief summary of the developmentprocess and resulting model statistics, as well as scores across different populations measured sofar (3 institutions, variety of courses). The student responses include new data since thepublication of the instrument as well as additional analysis of groups, in particular a comparisonof attitudes between genders that matches with recent results relating STEM persistence toattitudes and beliefs (Ellis et al., 2016; Wang et al., 2016).Key words:
2017
San Diego, California
A study was conducted in a small, private university in Northeastern United States inorder to determine if introducing an online component to a first year Calculus course wouldinfluence student learning. An online component was presented in two sections of the calculuscourses while two sections were taught using the traditional format. Preliminary data suggest apositive correlation between the online component and improved student performance in thecourse.Keywords:
2017
San Diego, California
Engineering educators are challenged with students at greatly varying mathematical skill levelswhile needing to quickly bring all students up to the same mathematical mastery level atappropriate points during a semester. To address this problem our team designed a teaching e-tool in WeBWorK called Just-In-Time Assessment and Review (JITAR) to be delivered as an on-line system consisting of a series of individualized mathematics modules inserted withinengineering courses at strategic points in the semester, prior to students needing those mathskills. JITAR assesses the mathematical competency level of the individual student and providesformative individualized learning opportunities in time for the students to be successful inapplying the necessary mathematics to the new engineering material. The new type of WeBWorKassignment was designed to support the desired presentation and flow of the module integratingassessment and e-learning assistance by offering a customized learning path to students. Thisproject is currently funded by National Science Foundation.Key words:
2017
San Diego, California
This poster presents a framework for characterizing teachers’ decentering during teacher-student interactions when teaching. Analysis of video data of a graduate teaching assistant’s(GTA’s) precalculus class generated six levels of teacher-student interactions. These levels willbe described and illustrated with excerpts from this video analysis.Key words:
2017
San Diego, California
In this communication, we will present various arguments supporting the claim that it isworthwhile taking in account logical issues in research in undergraduate mathematicseducation. We will provide arguments relying on research on student’s difficulties and their linkswith teachers’ practices on the one hand; on relevance for researchers on the other hand.Key words:
2017
San Diego, California
This poster describes and includes a discussion of the learning benefits and gains that students,enrolled in a Calculus I course, and their instructor reported as a result of participating in anactive learning environment. The alignment between what the instructor valued and what 67students experienced as they engaged in Calculus I activities was assessed using a survey. Theresults indicate a positive relationship between perceived importance and reported frequency ofengagement, which resulted in benefits and student learning. Opportunities for improvementoutcomes were found and can serve to support strategies that improve teaching and learning inCalculus I.Key words: Calculus I, Engagement, Learner-centered Instruction
2017
San Diego, California
This study considers calculus students’ conceptualization of average rate of change at a private liberal arts college in the Midwest. Researchers have indicated that undergraduate students do not develop productive meanings for average rate of change. In order to explore undergraduate students’ meanings for average rate of change further, we conducted clinical interviews with 10 undergraduate students on a four-item test. Participants were undergraduate students taking Calculus 1 at the time of the study. Interviews were conducted towards the end of the semester to ensure students have learned average rate of change. Qualitative techniques were used to analyze data. We will present and interpret data highlighting the techniques used by the students during the tasks. We will conclude with implications from our findings and questions for future research.
2017
San Diego, California
For the past three years we have run a seminar for 60 – 75 women in RUME the day before the annual conference called MPWR: Mentoring and Partnerships for Women in RUME. Participants included graduate students, post-doctoral fellows, faculty, and researchers outside of academic positions. In this poster, we provide a window into these seminars, specifically addressing the motivation for the seminar, the structure of the seminar, topics discussed in the seminar, research related to the efficacy and transferability of MPWR, and the future of MPWR. Our hope with this poster is to both share what we have been doing and get feedback from the community for what more can be done.
2017
San Diego, California
The Mathematical Problem Solving Item Development Project is designing and developingLikert items that capture students’ capacity in mathematical problem solving (MPS). The project,now in year two, continues to refine items that capture aspects of MPS. The refinement processincluded piloting items on over 1000 students in College Algebra and Calculus, hour-long think-aloud interviews with 26 students, and review by experts. The goal of this poster presentation isto provide information about the item development and design and gather feedback andsuggestions on further design and development.Key words
2017
San Diego, California
In this poster we report on results from the Group Concept Inventory (GCI), a conceptualassessment for introductory group theory students. Over 400 students from thirty institutions tookthe inventory. We use the framework of reducing abstraction (Hazzan, 1999) to situate studentresponses. We found that students frequently reduced abstraction (in a multitude of ways) whendealing with fundamental concepts in group theory.Keywords: Abstract algebra, Reducing abstraction, Concept inventory
2017
San Diego, California
In this presentation, I discuss the viability of a mathematical game as a learning tool for abstract algebra—specifically, the groups of order four. Throughout 2016, I designed and tested variants of my group theory card game, Groups, among individuals ranging from no post-secondary mathematics experience to current or prior graduate level mathematics study. Here, I review the design choices and challenges central to working in a game space drawing heavily on abstract algebra, and assess alterations to the game's mechanics influenced by my interactions with players.
2017
San Diego, California
This study, involving 254 college-level calculus students and 3 teachers, investigated the misunderstanding of concepts in calculus and designed concept-based instruction to help students understand concepts. Multiple achievement measures were used to determine the degree to which students from different instructional environments had mastered the concepts and the procedures. The midterm examination and the final examination results showed that the students enrolled in the concept-based learning environment scored higher than the students enrolled in the traditional learning environment and the investigation at the end of the semester showed that most of students like the concept-based learning environment.
2017
San Diego, California
There is an overwhelming amount of evidence that the incorporation of active learning in theclassroom benefits all students and can be especially beneficial for women and underrepresentedpopulations. However, our work is not finished when it becomes an integral part of teaching andlearning across the nation. Classroom settings that foster group interaction and collaborationmay result in an environment that is even more undermining to underrepresented populations. Inthis poster we illustrate these potential issues that arose in an abstract algebra course.Key Words:
2017
San Diego, California
The research community shares a concern for students’ conceptual understanding of calculus and commonly advocates for student-centered approaches as a way to promote it. In this study, we investigated the effect of different instructional approaches on 151 undergraduate students’ conceptual understanding of differential calculus in context-specific, natural settings. We collected data on the pre- and posttest of the Calculus Concept Inventory in three classes. In one class, most of the time was dedicated to conceptually oriented problem solving, another class implemented practice problems for students, and the third class was a traditional lecture class. The results showed that there was no difference in students’ conceptual understanding of differential calculus controlling for their initial understanding. Thus, our findings do not support the research that advocates for student-centered instruction suggesting that the approaches’ implementation and contextual differences may be sources of variation in their effectiveness.
2017
San Diego, California
The mathematical content presented during instruction has been shown to have an effect on studentachievement. To investigate the content presented by instructors during College Algebrainstruction, the Mathematical Quality of Instruction (MQI) observation protocol was applied tovideo recordings featuring instructors’ presentations of examples of solving quadraticinequalities. Wide variation was observed in the solution methods chosen by instructors and in therationale provided for choosing a particular procedure. This poster summarizes the variation inthe mathematics that was observed and the ability of the MQI protocol to capture this variation.Key words:
2017
San Diego, California
2017
San Diego, California
The Improving Undergraduate STEM Education Through Adjunct Mathematics InstructorResources and Support (IUSE-AMIRS) project aims to measure the impact of coursecoordination and support on adjunct mathematics instructors’ knowledge, instructionalpractices, and job satisfaction. In this project, we use the organization and coordination ofPrecalculus with the goals of 1) implementing best practices for learning and instruction, 2)improving instructor knowledge, and 3) creating a professional learning community. As a part ofthis project we measure the impact of Precalculus course coordination and adjunct support onstudent achievement, leading to student retention in STEM majors. We believe our initiative canbe implemented in other departments and institutions that have a similar need for adjunctinstructors in math courses with multiple sections.Key words:
2017
San Diego, California
Given the recent national and international events the need for developing students’ quantitativeliteracy (QL) has taken center stage in the mathematics education community. We are interestedin investigating the existing support structures and the impact they have on the development ofQL. The purpose of this study is to investigate the literature on quantitative learning centers atinstitutions of higher education. This poster will discuss the themes that emerged from aqualitative analysis of these works, highlighting what we currently understand and identifyingopportunities for growth.Key words:
2017
San Diego, California
Mathematical problem-solving research studies abound, and a significant portion express therole of metacognition as an underlying component of the problem-solving process.Unfortunately, much of the research on metacognition in mathematics does not describethe explicit role metacognition plays during the problem-solving process. Moreover,metacognitive interventions are typically disconnected from natural mathematical activity anddiscourse within a classroom community. The purpose of this qualitative study is to characterizesociomathematical metacognitive norms within an introductory number theory course intendedfor pre-service teachers
2017
San Diego, California
The terms equity, diversity, inclusion, and social justice have entered the research lexicon. Thistheoretically-focused poster presents some recent policy efforts to generate a shared meaning for“social justice” in mathematics education and offers a theoretical framework for making senseof (and making sense with) intercultural interactions as an essential component of rigorousresearch. The poster includes illustrations for how to use these tools for thinking through andtalking about research. To anchor discussion, we focus on research on teaching and learning inthe courses before calculus (e.g., algebra, mathematics for pre-service elementary teachers).
2017
San Diego, California
Studies were conducted to explore the efficacy of a non-traditional transition to upper division proof course using Freudenthal’s notion of mathematizing as a framework. Classroom video data and student work were analyzed using grounded theory methodology. Results indicated that students in the non-traditional course developed better understandings of the role of definition and counter example in proof through engagement in meaning making activities that fostered both the semantic and structural aspects of proof writing.
2017
San Diego, California
The goals of undergraduate mathematics teacher education include developing teachers’ contentknowledge and pedagogical content knowledge. As a strategy for conceptualizing and assessingthese forms of knowledge, researchers have further divided these domains. However, it hasproven difficult for research groups to create tasks to reliably capture a specific domain withoutinvolving other domains, leading them to question these subdomains. We argue that tasks’inability to measure subdomains separately is not evidence that tasks or theory are flawed.Instead, we propose that assessment tasks are effective in measuring MKT when they representthe work of teaching, rather than when they isolate subdomains. To make this argument, we usean analysis of teachers’ thinking in response to nine MKT assessment tasks. Though prior workprovides evidence that the tasks measure MKT, the tasks cannot be meaningfully parsed into thesubdomains of multiple established MKT frameworks.Key words:
2017
San Diego, California
The results of educational research studies are only as accurate as the data used to producethem. Drawing on experiences conducting large-scale efficacy studies of classroom-basedalgebra interventions for community college and middle school students, I am developingpractice-based data cleaning procedures to support scholars in conducting rigorous research.The poster identifies common sources of data errors in mathematics education research andoffers a framework and related data cleaning process designed to address these errors. I seekfeedback on the framework and discussion around data cleaning techniques used by otherRUME scholars in their research and in the preparation of future researchers.Key words:
2017
San Diego, California
Student engagement has been identified as a critical element in student learning ofmathematics, yet most university math classrooms have very little active content (Olson &Riordan, 2012). Guided by Schoenfeld’s (2011) framework for instructional decision-making, this study examines calculus instructor beliefs about, purposes for, and barriersagainst student engagement. Results indicate instructors utilize active learning primarilyfor formative assessment and improving student dispositions with development ofunderstanding as a secondary, implicit goal.Key Words:
2017
San Diego, California
This study explores a fundamental calculus connection between a function and its derivative byexamining and categorizing strategies students use when matching a function’s graph with thegraph of its derivative. Through interviews with four students using multiple choice (MC) tasks,we wanted to explore whether common mistakes and students’ strategies when drawing thegraph of the derivative of an original function are consistent with those found when usingopen-ended tasks. While tendency to find an equation of the graph in order to differentiate wasobserved, simple replication of the original function was not observed.Keywords:
2017
San Diego, California
Instructional practice and decision-making are influenced by a myriad of factors, with bothindividual instructor characteristics (e.g., beliefs about teaching and learning and personalexperience) and departmental/institutional characteristics (e.g., resources and supports) shapingday-to-day teaching practices. However, little is known about which factors are the mostinfluential and how those factors influence pedagogy. In this project, in an effort to identifycommonalities in situational contexts and better understand how these commonalities supportnon-lecture instructional approaches, we look at interviews from fourteen mathematicians whovolunteered to implement inquiry-oriented instructional materials.Key words:
2017
San Diego, California
Students are frequently asked to reason about graphs that they see as geometric shapes, insteadof representations that show the relationship between two quantities. This study shows aninstructional intervention, using the theory of multiplicative objects (Saldanha and Thompson,1998) that has great potential for orienting students to the quantities involved and theirrelationships, by focusing on how graphs display orthogonal lengths whose magnitudes aremeasures of quantities.Key words:
2017
San Diego, California
In introductory physics classes, student frequently experience difficulties with relative motionproblems. Previous studies have categorized student difficulties with reference frames, or usedcomputer simulations or experiments to create seeming paradoxes that students would needframes of reference to resolve; however, these studies failed to define what they meant by a“frame of reference” in the mind of a student. In 2016 I carried out a pilot study that used ourcognitive definition of a conceptualized and coordinate frame of reference (Joshua, Musgrave,Hatfield, & Thompson, 2015) as well as quantitative reasoning (Thompson, 1993) to guide aninstructional intervention and analyze the difficulties the student had with relative motion tasks.Both constructs proved to have great explanatory power, as they revealed aspects of thestudent’s thinking that were not commonly explored in previous studies. Both the results of thisstudy and their implications will be the topic of my poster.
2017
San Diego, California
This project attempts to seek out common threads and analyze discrepancies in the tertiary-levelmathematical literacy / quantitative literacy curricula proposed by eight different textbooks andcontent-providers. Following the framework developed originally in (Harel 1987) we investigatesequencing of content, levels of generality, emphasized applications, introductory material, aswell as explicitly stated learning outcomes.Key words:
2017
San Diego, California
An “Introduction to Proof” or “Transition to Proof” course is widely offered as an essentialpart of the undergraduate mathematics curriculum at most post-secondary institutions. Thisposter reports on the iterative development of one such course that used an Inquiry-basedapproach to the teaching and learning of mathematical proving and proof. Moreover, thechanging beliefs of students, about the nature of mathematics and about doing mathematics, inthis course, are discussed.Keywords:
2017
San Diego, California
Using evidenced-based practices in a large undergraduate mathematics classroom can bechallenging but results of recent research on active-learning demand investigation. Preliminaryresults show that students already have self-confidence upon entering the classes, but there is aslight gain in perceptions of the value and appreciation of mathematics. Additionally, activitiesand clickers were considered useful and important by some of the students interviewed.Keywords
2017
San Diego, California
One of the reasons for the exodus in STEM majors is students’ experiences in their first undergraduate mathematics course, usually introductory calculus. However, students with high calibration are more likely to be aware of their deficiencies and seek assistance in time for it to be effective. Although there is evidence that students who regularly complete post class reflections are more successful than those that do not, it is not known if such assignment also improves students’ calibration. The purpose of this correlational study was to investigate to what extent students enroll in CLEAR calculus become more growth mindset orientated the relationship between post-class reflections, calibration, and achievement in introductory calculus.
2017
San Diego, California
We report on an exploratory quantitative study of students’ error detection skills. Based on theresearch on “proof framework” type errors and models for proof comprehension, we classifyerrors in proofs as internal or external for proof validation. We then test students' ability todetect errors of both types and determine if detection is correlated with success in anintroductory proofs course.Key words:
2017
San Diego, California
At Utah State University, the course known as ‘Math 4200:Foundations of Analysis’ is a requirement for all department majors, and, in addition to introducing real analysis, serves as an introduction to rigorous proof. All too frequently, courses such as these are taught with the typical lecture format: the instructor enters the classroom to deliver polished explanations of definitions, theorems, and their proofs, while leaving students to struggle to follow lectures and then struggle further on their own to make sense of incomprehensible homework problems. This poster includes descriptions of easy-to-implement strategies that change the classroom to an active learning environment, with ample opportunities for formative assessment and feedback without overloading the professor with busywork. The strategies include: name tents, group exercises and quizzes, peer-reviewing of homework, concept quizzes, individual presentations, and growth mindset reflections. We surveyed students to find their reactions to these class activities and found positive and helpful implementation hints.
2017
San Diego, California
This exploratory study investigates pre-service teachers’ (PSTs) collaborative discus-sions to solve probabilistic problem. The PSTs synchronously collaborated online using Virtual Math Teams with GeoGebra to investigate the fairness of a series of die by using interactive simulation to randomly sample from the die with replacement. While discussing their solution in the chat panel, the PSTs used informal, non-standard language to describe the distribution of the data. In this poster, we present examples of how the PSTs, using informal language, co- constructed their knowledge of different probabilistic concepts while solving the problem. This study contributes one of a series of tasks that were designed to elicit how PSTs build understandings of mathematical concepts without formal introductions to these concepts.
2017
San Diego, California
In this research presentation I utilize the theoretical perspective Knowledge In Pieces (diSessa,1993) to identify the knowledge resources two students utilized while in the process ofcompleting various differential equations tasks. The results provide a fine-grained description ofthe knowledge students consider to be productive with regard to completing various differentialequations tasks. Further the analysis resulted in the identification of five ways students framedifferential equations tasks and how these framings are related to the different knowledgeresources students utilize while completing the various tasks. These framings did not onlyprovide insight into the students’ general approaches to completing the task; differences in theindividual students’ applications of knowledge across the tasks were accounted for by thedifferent Framings. The results have direct implications with regard to the teaching ofdifferential equations as they inform the ideas students view as productive when completingvarious tasks involving differential equations.Keywords: Differential Equations, Knowledge in Pieces, Undergraduate Student Learning
2017
San Diego, California
We present a graphic review of a decade of research and evaluation work on inquiry-basedlearning in college mathematics. Featured studies examine student outcomes of IBL instruction,processes of change as instructors explore and adopt IBL approaches, and the workings of thefaculty learning community that has formed to learn and promote IBL ideas. Collectively, thesestudies highlight the use of evidence to understand instructional practices and change in thesepractices, and to support evidence-based decision-making by instructors and change leaders.Key words:
2017
San Diego, California
2017
San Diego, California
Students encounter multiple mathematical representations of change in physicscourses. In addition to the complexity of the material, students must navigate mathematicalnotation that can seem arbitrary and can differ from conventions used in mathematicscoursework. In this poster we will examine student responses illustrating the challenges ofmathematical representations of change, drawn from students in upper-division physicscourses in mathematical methods and thermal physics.
2017
San Diego, California
Developmental – or the antiquated “remedial” – mathematics is a large enterprise in Americancolleges. For the California State University (CSU) system roughly one-third of all studentsrequire developmental mathematics. Placement in these courses in the CSU is determined by astandardized test. Those who fail are required to take some campus-specific variant ofdevelopmental mathematics. This poster addresses the question, what more can be said aboutstudents enrolled in developmental mathematics programs other than they have failed an exam?An analysis of survey instrument data will be presented that shows San Jose State Universitydevelopmental mathematics students are fundamentally different, undesirably so, than their non-developmental counterparts on a range of attitudinal, affective, and dispositional measures.Key words:
2017
San Diego, California
Mathematics textbooks and mathematics education research articles frame countingproblems as requiring clever insight and being inherently challenging and especially accessible.In this study, we distributed a survey to mathematics students in order to examine studentattitudes about counting problems and the extent to which these attitudes aligned withpresentations of counting in the literature. In this poster, we present results from this survey thathighlight some surprising ways in which the responses did and did not align with the literature.
2017
San Diego, California
We present the research design and data collection strategies for a federally funded project(Watkins, Duranczyk, Mesa, Ström, & Kohli, 2016) that investigates the connection betweeninstruction and student learning and performance in algebra courses at community colleges. Theposter focuses on measurement issues we face in identifying the characteristics of mathematicsinstruction and students’ learning gain, specifically we address questions encountered from thepilot data collection (six community college faculty and nearly 150 students) that need to beresolved prior to data collection.Key words:
2017
San Diego, California
Creating a culturally inclusive classroom has been suggested to help minority students improvetheir achievement in class. However, evidence shows gaps between teachers and students aboutwhat a culturally inclusive classroom should be. We propose a framework for investigating thedifferences between teachers’ and students’ beliefs on such a classroom.Key words
2017
San Diego, California
In 2010, Charalambous published an article that examined the relationship betweenmathematical knowledge for teaching (MKT) and task unfolding. As a result of this study,Charalambous found evidence to support the claim that there is a positive relationship between ateacher’s MKT and the cognitive level of task presentation and enactment. Drawing upon thisfinding, the purpose of this case study is to utilize unfolding and cognitive demand as a lensthrough which to examine mathematical knowledge for teaching at the undergraduate level.While MKT has been studied extensively at the K-12 level, there are relatively few studies thatfocus on MKT at the collegiate level. In order to help fill this gap, this case study first identifieshow Precalculus instructors unfold examples that involve procedures and then examines theMKT that is involved in this unfolding.Keywords:
2017
San Diego, California
250 calculus students were asked to evaluate f′(2) when f(x) = x3 if x = 2 and f(x) = 8 if / x = 2. Responses were coded, and eight students were interviewed about their answers. The data provide insight into students’ understandings of function, derivative, and graph.
2017
San Diego, California
Researchers have reported on the difficulties K-12 students, pre-service and in-service teachersexperience in reasoning and communicating about angle, angle measure and trigonometricfunctions. This work extends the existing literature to highlight that even mathematicallysophisticated individuals (e.g., PhD mathematics students) often struggle to speak with meaningabout these ideas sans targeted interventions to support them in doing so. We share tasks anddata from semi-structured clinical interviews conducted with graduate teaching assistants tohighlight differences in communication about these ideas pre- versus post-intervention.Keywords:
2017
San Diego, California
Part of the work of mathematics teacher educators (MTEs) are to provide authentic experiences to prospective teachers. We showed four temperature story problems involving integers to prospective teachers (PTs), and asked them if the stories matched given number sentences. While each of the stories were similar to the number sentence, none of them matched exactly. We examine the reasons PTs gave for saying that the stories matched the number sentences, and discuss implications of their thinking for mathematics content courses for prospective teachers.
2017
San Diego, California
This study explores classroom participation as an agent of socialization for preservice elementary mathematics teachers. Each character, the teacher, student and curriculum in the classroom plays various roles in the socialization process. However, the teacher plays a major role because s/he is responsible for creating the environment where participation is possible. In this work, the analysis of data shows the teacher’s teaching method(s), questioning and listening skills, as well as her background understanding of the specialized mathematics knowledge needed for teaching and the students all help to create such an environment. Also, as students explore mathematics by doing through group work, class discussions and individual work, they experience growth in mathematics classroom practices that results in change in the mathematics identity of the students.
2017
San Diego, California
We have a collection of ongoing studies designed to investigate the impact of Team-Based Learning (TBL) in calculus instruction on student learning. The first study involves the implementation of TBL in Calculus I and II. Initial findings suggest that TBL students have larger score gains on the Calculus Concept Inventory (CCI) than students receiving traditional instruction. However, there seems to be a gender gap as women tended to have smaller CCI gains than men. The second and third studies investigate the transfer of calculus to major courses, one by asking calculus content questions in subsequent major courses and the other through the educational setting of first-year student Learning Communities.
2017
San Diego, California
As part of a larger study, we analyzed focus groups of students discussing their perceptions of anexperimental real analysis course. The aim of this course was to teach real analysis toprospective and practicing teachers in a way that improved their future teaching. This posteranalyzes data from four focus group interviews from 20 students after they completed theexperimental course. The majority of comments from the participants’ comments about thecourse, both in general and in regards to informing their future teaching of secondarymathematics, were favorable. We present commonalities in the participants’ responses.Keywords:
2017
San Diego, California
Extreme Apprenticeship is a novel, student-centred teaching method that is designed for teachinglarge courses with hundreds of students. It is based on Cognitive Apprenticeship. In this poster,we present the Extreme Apprenticeship method and data collected from courses taught with it.Key words:
2017
San Diego, California
Part of a larger study of the development of teaching among novice college mathematics instructors, this report focuses on one participant, Disha, and her use of a questioning technique called hypophora. At the beginning of the observations, 25% of her questions were hypophora. After video-case based activities during weekly coordination meetings her use of hypophora decreased to about 10% of questions. Although Disha rejected the idea that her teaching had changed in any way, she acknowledged that she began “breaking things into smaller pieces” to help students understand.
2017
San Diego, California
We explore understanding of the Existence and Uniqueness Theorems (EUTs) by a group of engineering students working on nonstandard problems. Students presented three sets of solutions: individual solutions produced in the first tutorial, individual solutions submitted as a homework, and solutions submitted after the discussion with peers in small groups during the second tutorial. The focus of the study is on the role of individual and group work with nonstandard problems. The results show that students gained a deeper understanding of EUTs and appreciated the experience.
2017
San Diego, California
The formal definition of the limit of a function was taught in a first-year calculus course usingopen intervals and a topological approach. Student understanding was supported with computer-based visualization tools. The concept image framework was used to interpret results of a pilotstudy in which data was gathered through concept maps and analyzed using categorical content
2017
San Diego, California
Quantitative and Co-variational reasoning have been shown to be important facets of a student’smathematical learning. We are proposing an online workbook as a tool for supporting students inreasoning quantitatively and co-variationally. In this poster, we will briefly present a section onunderstanding graphs as representing the co-variation of two quantities’ values.
2017
San Diego, California
Engaging in mathematical problem posing activities can have positive effects on students’mathematical thinking and can advance students’ understanding of mathematical concepts.Knowing how underprepared undergraduate students pose problems informs the use of problem-posing activities for helping these students advance their understanding of mathematics as theytransition to college-level mathematics courses. Forty-five undergraduate students enrolled in adevelopmental mathematics course participated in a written problem-posing assessment todescribe what underprepared undergraduate students’ problem posing looks like. Students’written responses were assessed for whether the response was a mathematical question, whetherthe responses were solvable, and the connections between each response a student provided.Results of the assessment indicate students at all levels of course performance posed solvablemathematical problems and commonly posed problems by changing the objective for each problemcreated.Key words:
2017
San Diego, California
As part of a proof-of-concept project, we created multi-media activities and instructor supportmaterials for secondary mathematics teacher preparation. One focal topic was transformationalgeometry. Data collection included undergraduate and secondary school students responding totasks in surveys and in interviews. Despite its prominence in the Common Core State Standardsfor Mathematics, little is known about how students think about ideas in transformationalgeometry or about how they engage with items used on assessments for this topic. The posterreports findings on student thinking and invites discussion to inform future work.Keywords:
2017
San Diego, California
The aim of this study is to investigate students’ transition between the three worlds of mathematical thinking and the challenges that they face in making these transitions. We anticipate that by creating more opportunities to move between the worlds we will encourage students to think in multiple modes of thinking and hence gain richer conceptual understanding.
2017
San Diego, California
In this talk we present a theoretical framework based on Skemp’s idea of schema. According to Skemp, concepts are embedded in a hierarchical structure of other concepts, these levels in the structure being classifications of concepts. As the concepts are paired together, relations between them as well as classifications are also possible. The complexity of this hierarchical structure comes from the fact that these classifications of concepts and relations are not unique, giving way to multiple hierarchical structures, which can be interrelated. When components of these conceptual structures come together to make a structure that would not be realized by only looking at the individual components, the resulting structure is called a schema.
2017
San Diego, California
Instructors often want to evaluate their students’ degrees of conceptual understanding in theirmathematics courses, but are typically limited to course assignments and exams. In this researchwe ask: To what extent can mathematics instructors recognize conceptual understanding of theirstudents based on final exam responses? During a summer REU program we examinedpre-existing exams along with other course materials to address this question. We developedcodes for student responses that were guided by Anderson and Krathwohl (2001), Mejia-Ramoset al. (2011), Thurston (1994), and the APOS framework. Student responses more clearly andoften demonstrated lower level understanding than deeper, conceptual knowledge because fewproblems called for explanations or justifications. The goal of this research was to improve theeffectiveness of assessments in evaluating conceptual understanding. In future exams, we suggestthat prompting students to display behaviors typical of varying levels of understanding withjustification would make evaluations more accurate.Keywords:
2017
San Diego, California
The CCSSM emphasize statistical concepts for grades 6-12. The paper will attempt to answer the following question: How do middle school mathematics teachers in a professional development program differ from each other with regard to how they view statistics?
2017
San Diego, California
We present preliminary findings from written, post-instruction surveys to gauge studentunderstanding of various elements of multivariable calculus. The content addressed includescontour plots, partial derivatives, representations of gradients and slopes, construction ofvolume difference integrals.Key words:
2017
San Diego, California
The study presented here is an illustrative example of an action based research project, which was focused on broadening student partition in the flipped classroom experience in order to address issues of equity and social justice in the calculus curriculum. While flipped classrooms have gained recent notoriety within the literature, they rarely incorporate or addresses other critical perspectives. Our study highlights how using design principles such as realistic mathematics education (RME) and culturally responsive pedagogy can effectively target the hidden curriculum and shape the norms and classroom discourse features (Sfard, 2008).
2017
San Diego, California
The study presented here examines the types and relative frequency of uniform course components (exams, textbooks, etc.) currently in place in the Precalculus through single variable calculus sequence at graduate universities and how those components are effected by the presence of department factors such as regular course meetings, instructor type, and the presence of a course coordinator. Our results indicate that while the total number of uniform course components decline throughout the Precalculus through single variable calculus sequence, its effect is mitigated by the presence of a course coordinator and regular course meetings. In addition, student success is significantly related to the presence of both a course coordinator and regular course meetings.
2017
San Diego, California
It is well documented that the precalculus through single-variable calculus sequence (P2C2) actsas a barrier for many STEM intending students. Students often cite poor instruction as a primaryreason for switching out of STEM programs (PCAST, 2012; Seymour & Hewitt, 1997), whichleads to questions about what instructors and instruction look like across the country. Thisposter presents findings from national census survey data collected as part of a larger study,Progress through Calculus (PtC). In particular, we answer: (1) What types of instructors arecurrently teaching courses in the P2C2 sequence and how prevalent are they nationally? (2)What relationship exists (if any) between instructor type and primary instructional format?
2017
San Diego, California
Graphing calculators have been used for teaching introductory statistics for decades. They helped students to obtain accurate statistical analysis results. However, heavily relying on graphing calculators may hinder students’ understanding of certain statistical concepts such as the normal distribution and p-value. In this study, we focused on the effects of using a graphing calculator on students’ conceptual understanding of normal distribution and p-value, and their performance of calculating normal probabilities and conducting a hypothesis test.
2017
San Diego, California
This study explores beliefs about doing math held by pre-service teachers. Pre-service teachersin a mathematics content course drew pictures of a person doing math. Additionally, modifiedFennema-Sherman Mathematics Attitude Scales (FSMAS) (Ren, Green, & Smith, 2016) wereadministered to the students. The drawings were analyzed using a framework developed from theFarland-Smith (2012) rubric. In addition to exploring beliefs about doing math held by thestudents as evidenced by the drawings, the study considers the validity of the drawingmethodology through a comparison to the FSMAS results.Key words:
2017
San Diego, California
Mathematics faculty and education researchers increasingly recognize the value of the history of mathematics as a support to student learning. There is an expanding body of literature in this area which includes direct calls for the use of primary historical sources in teaching mathematics. The current lack of classroom-ready materials poses an obstacle to the incorporation of history into the mathematics classroom. Transforming Instruction in Undergraduate Mathematics via Primary History Sources (TRIUMPHS) is a seven-institution collaboration that will design, implement, test, and publish curricular materials based on primary historical sources, train approximately 70 faculty and graduate students on their development or implementation, and conduct and evaluation-with-research study. We present an overview of the project, including activities and research to date.
2017
San Diego, California
Using survey data and interviews from a large urban university system, this study explores factorsthat impact student decisions to take math classes online. The results suggest that access to onlinemath courses likely impacts student course taking patterns, with significantly more students takinga different course if their desired math course is not offered online, compared to non-math courses.Keywords:
2017
San Diego, California
During academic service-learning experiment, students in an experimental Precalculus classregularly tutored basic algebra to middle-schoolers. At the end of the quarter, student-tutorsdemonstrated academic improvement and a shift in beliefs about importance of conceptualunderstanding in problem solving. These manifested benefits can motivate mathematicsdepartments to implement service-learning as part of academic curriculum.Key words:
2017
San Diego, California
Successful proof production in advanced mathematics relies on meaningful apprehension ofthe to-be-proved proposition, yet we know that this is a challenge for many students. Thisstudy examines the ways in which a mathematician-instructor, addressing this issue, modelsthe practice of interpretive reading of mathematical propositions to a pair of students in thecontext of joint proof-production in a Real Analysis course. Taking a social practiceperspective on reading and adopting Sfard’s commognitive framework as a theoretical lens, Iidentify three aspects of the discursive work the expert engages with to demonstrateprocesses of active meaning-making to students: (1) re-reading of text with grammaticalshifts, (2) posing comprehension monitoring questions, and (3) narrative enactment of text
2017
San Diego, California
The concept knowledge and attitudes and perceptions about mathematics of students inprecalculus and calculus were measured using the Precalculus Concept Assessment (PCA) andMathematical Attitudes and Perception Survey (MAPS). We found significant differences in thesemeasures among several subgroups, and found correlations between these measures and studentsuccess.