PROCEEDINGS OF THE 23RD ANNUAL CONFERENCE ON RESEARCH IN UNDERGRADUATE MATHEMATICS EDUCATION
2020
Boston, Massachusetts
Number of pages: 1291
Number of papers: 213
- Shiv Smith Karunakaran
- Zackery Reed
- Abigail Higgins
2020
Boston, Massachusetts
Number of pages: 1291
Number of papers: 213
2020
Boston, Massachusetts
Using engaging problem contexts is important in instruction, and the literature contains themes of contexts being realistic, worthwhile, or enjoyable, as well as motivating. Yet, the literature largely lacks detailed student perspectives on what helps problem contexts achieve these characteristics. In this study, eleven calculus students were interviewed to identify features of problems that made them engaging. This led to a new top-level characteristic “variety,” and the identification of features that helped contexts have the characteristics described in the literature. In particular, problems that were realistic/motivating contained features including: (a) expansion of awareness, (b) need for math, and/or (c) explicit purpose. Contexts that were enjoyable/motivating contained features including: (a) insertion into problem, (b) teacher’s personal story, or (c) absurd story. At the end, we show the usefulness of these results by critiquing problems from the literature in terms of how engaging they might be to students.
2020
Boston, Massachusetts
The Mathematical Education of Teachers as an Application of Undergraduate Mathematics project provides lessons integrated into various mathematics major courses that incorporate mathematics teaching connections as a legitimate application area of undergraduate mathematics. One feature of the lessons involves posing tasks that require undergraduates to interpret or analyze the work of another student. This paper reports on thematic analysis of hour-long interviews for eight participants enrolled in an undergraduate abstract algebra course from two different implementation sites. We focus on student work and reactions to these interpreting or analyzing student thinking (AST) applications as they relate to their perceptions regarding the use of AST applications as a mechanism to both deepen their content knowledge and improve their skills for communicating mathematics. Several participants identify positive benefits, but more research is needed to determine the how to incorporate AST applications to accommodate some participants’ reluctance to engage in new mathematical contexts.
2020
Boston, Massachusetts
For decades, the mathematics education research community has strived to identify, document, and disseminate instructional strategies that support student success in mathematics courses. We now have many research-based instructional strategies (RBIS) which have evidence to support claims that they can be used to help students succeed in mathematics. However, the use of such practices has not become widespread at the postsecondary level in mathematics, nor in the other STEM disciplines. This work, focused on mathematics instructors, is part of a larger project hoping to uncover more about why and how particular instructional practices spread in undergraduate mathematics, chemistry, and physics. Here, we report that single-variable calculus instructors have awareness of, and use, RBIS in their classrooms; however much of class time is spent in a lecture format. Implications of these and future results are discussed.
2020
Boston, Massachusetts
The purpose of this study is to investigate the visual and nonvisual approaches and conceptualizations of slope used by introductory calculus students when presented with a variety of tasks.
2020
Boston, Massachusetts
For over thirty years, the mathematics education research community has investigated individuals’ function conceptions and advocated the importance of a strong function conception. This is particularly important in the case of preservice secondary mathematics teachers (PSMTs) as their knowledge of function will influence their future teaching of function. This study explores how one PSMTs’, Sofia’s, function concept image changes when she engages with research-based tasks and explorations designed to elicit cognitive conflicts related to function conceptions. Thematic analysis methods were applied to open-ended, tasked-based pre- and post-interviews designed to evoke aspects of individuals’ function concept image. Between the pre- and post-interviews, Sofia participated in a course that implemented 11 explorations on functions. Comparing the themes identified in her pre- and post-interviews, researchers identified four changes in Sofia’s function concept image. These changes relate to beliefs about functions and equations, non-numerical functions, functions and graphs, and the vertical line test.
2020
Boston, Massachusetts
In this presentation I will describe the beginning phases of a local instructional theory that has resulted from a design experiment focused on students work with symmetry groups in the context of chemistry. This local instructional theory will consist of both a generalized instructional sequence intended to support the guided reinvention of a classification algorithm for molecular structures, and the theoretical and empirical rationale for the given sequence. I will also describe the various teaching experiments and how each informed the development of the instructional sequence. The mathematical activity of the participants will be used to describe key aspects of the reinvention process.
2020
Boston, Massachusetts
Active learning has been shown to increase student success and improve student confidence, whereas ambitious teaching potentially leads to decreased student attitudes for learning mathematics. In this study, we examined how 18 Calculus I students’ beliefs, attitudes, and expectations were met or challenged across a semester when taught using Team-Based Learning, an ambitious teaching method. Results indicate that aspects of the TBL design, including initial interaction with content outside class, reduced emphasis on instructor lectures, and complexity of the tasks may have contributed to students’ negative attitudes. These negative attitudes interplayed with students’ beliefs about their competence to learn mathematics on their own and their beliefs about the nature of mathematics and how mathematics should be learned.
2020
Boston, Massachusetts
Future mathematics teachers must be able to interpret a wide range of mathematical statements, in particular conditional statements. Literature shows that even when students are familiar with conditional statements and equivalence to the contrapositive, identifying other equivalent and non-equivalent forms can be challenging. As a part of a larger grant to enhance and study prospective secondary teachers’ (PSTs’) mathematical knowledge for teaching proof, we analyzed data from 26 PSTs working on a task that required rewriting a conditional statement in several different forms and then determining those that were equivalent to the original statement. We identified three key strategies used to make sense of the various forms of conditional statements and to identify equivalent and non-equivalent forms: meaning making, comparing truth-values and comparing to known syntactic forms. The PSTs relied both on semantic meaning of the statements and on their formal logical knowledge to make their judgments.
2020
Boston, Massachusetts
Quantitative and qualitative evaluation of math tutoring centers is a critical step to identify characteristics of effective centers. A group of ten math tutoring centers gathered quantitative and qualitative measures of effectiveness as part of an ongoing project to identify characteristics of effective centers. This report summarizes the data collected. We will use this data in a future paper to generate testable hypotheses about characteristics of effective math tutoring centers.
2020
Boston, Massachusetts
In this paper we explore a wide sample of currently available instructional materials intended for college mathematics instructors (textbooks, magazines, teacher editions, lesson plans, teaching articles, classroom notes for flipped classrooms, books, etc.) in order to assess how available materials are building a knowledge base for teaching. We modify a framework from Hiebert & Morris (2009) to look for key categories of knowledge that are fundamental for a knowledge base for teaching mathematics. We found that few articles contained meaningful amounts of multiple categories. We use the categories to describe the nature of current available materials and argue that a new genre of instructional material and scholarly work to create the missing knowledge is needed
2020
Boston, Massachusetts
In this paper we use data from a recent teaching experiment to characterize a particular challenge in proof comprehension that arises when a proof uses the same definition multiple times in related ways. This difficulty involves coordinating the role and value of a quantified variable in the definition, a distinction we explicate in the paper. We do so by providing two proofs that exemplify the key difficulty and presenting an episode in which two students who were reading a proof together adopted alternate interpretations of a variable (one as value and the other as role), but were for a time unable to coordinate the two meanings. The primary contribution of the paper is to sensitize the research community to this challenge in a way that will help future research on students’ reading of mathematical proof.
2020
Boston, Massachusetts
Counting problems have been shown to be challenging for students to solve correctly, and one reason is that they can be difficult to verify (e.g., Eizenberg & Zaslavsky, 2004). The field would benefit from further investigations into the nature of students’ verification of counting problems. One possible avenue for verification is to have students engage with computational tools and environments. In this paper, we investigate how students verify solutions to combinatorial problems in the context of using Python computer programming. We show that students do not simply use a computed numerical answer when verifying, but they demonstrate a rich understanding between a computer program and a counting process. Further, we exhibit some affordances that using Python to verify solutions provides to the students.
2020
Boston, Massachusetts
After the country’s focus shifted from K-12 towards higher education, postsecondary schools found themselves under significant public, financial, and political pressure. To close the achievement gap and meet new standards of accountability, higher education institutions began looking for methods to increase student access and success. This quantitative study measured the effectiveness of implementing a critical statistics pedagogy in an undergraduate introductory statistics classroom and its impact on course success, persistence, and mathematical empowerment. Data collected from four classes at a community college found the use of a critical pedagogy had a positive impact on students’ overall achievement, increased their awareness of social justice issues, and aided in the development of their critical voice.
2020
Boston, Massachusetts
The Pathways to College Algebra curriculum aims to build concepts that cohere with the big ideas in Calculus, and initial results suggest improved readiness for Calculus by students who use the curriculum. Our study examines similarities and differences of Pathways and non-Pathways students understanding and reasoning about the calculus concept of the limit. We compare students’ understanding of limits at the beginning and at the end of the unit. Our findings suggest that (1) students reliance on procedures, combined, or quantitative reasoning was dependent on the calculus instructors’ emphasis in the class; (2) students who begin their Calculus class with high covariational reasoning gain a more sophisticated understanding of limits; and (3) when curriculum is coherent students will identify mathematical connections.
2020
Boston, Massachusetts
Homework is one of students’ opportunities to learn mathematics, but we know little about what students learn from homework or how they learn it. Online homework platforms contain a variety of resources students might access. This paper explores how students used a ‘practice another version’ (PAV) feature. Findings indicate students used PAV problems for extra practice, as templates, to see the steps for solving a problem, to make sense of a solution method, to troubleshoot when their own method did not work, to see the form of an answer, to maximize their score, and to check that their method was on the right track. The primary uses were for sensemaking, to see steps, or to use the problem as a template. This work lays groundwork for future work characterizing what students learn from homework and how features such as PAV help (or hinder) their learning.
2020
Boston, Massachusetts
College algebra continues to be a barrier for college students (particularly non-STEM majors) to graduate. The rigid teaching methods for developmental mathematics courses continues to exacerbate this issue causing failure rates among students. In an age where colleges are recruiting more students of color, these students are often regulated to these courses and are struggling to successfully complete it in one attempt. With innovate teaching strategies, this study sought to address this problem. The purpose of study was to investigate the effects of a sequence of lessons grounded in the principles of culturally relevant pedagogy on students enrolled in a college algebra course at a historically Black college/university. In particular, the study examined students’ views about mathematics and its interaction with culture. Results indicated that students in this course, had positive views about this course, despite their views about mathematics coming in to the course.
2020
Boston, Massachusetts
This study examines ways in which preservice teachers use mathematics in a social justice context. This study examines preservice teachers’ conceptions of teaching mathematics using a social justice lens. Situated at a large, public, predominantly white institution in the southeast United States, where preservice teachers are not required to take a course on teaching diverse populations, participants were asked to respond to questions surrounding their experience with a mathematical social justice activity adapted from Gutstein (2005). Using the preservice teachers’ responses from pre- and post-surveys, researchers were able to compare initial conceptions of teaching for social justice to understandings after an activity involving social justice topics of world wealth and population disparity. Preliminary results show that preservice teachers’ attitudes shifted from naive/general notions of teaching mathematics for social justice to somewhat more concrete ideas.
2020
Boston, Massachusetts
This paper presents narratives of change at three university mathematics departments in response to administrative insistence on improvements in the face of pressure from engineering colleges. Specifically, these three departments were faced with the prospect of losing calculus teaching to another unit on campus and reacted to prevent that outcome. These three departments initiated change efforts including the recruitment of newly appointed leaders to oversee improvements, and all have successfully maintained control of calculus because of perceived program improvements. Drawing on sociological perspectives, we discuss the ways in which change was navigated and negotiated at each site. We found common characterizations of change leadership limited these descriptions of complex change processes, suggesting a need to refine such characterizations in ways that capture more nuanced situations.
2020
Boston, Massachusetts
Identifying patterns is an important part of mathematical investigation, but many students struggle to explain or justify their pattern-based generalizations or conjectures. These findings have led some researchers to argue for a de-emphasis on pattern-based activities, but others argue that empirical investigation can support the discovery of insight into a problem’s structure. We introduce a phenomenon we call empirical re-conceptualization, in which learners identify a conjecture based on an empirical pattern, and then re-interpret that conjecture from a structural perspective. We elaborate this construct by drawing on interview data from undergraduate calculus students and research mathematicians, providing a representative example of empirical re-conceptualization from each participant group. Our findings indicate that developing empirical results can foster subsequent insights, which can in turn lead to justification and proof.
2020
Boston, Massachusetts
Undergraduate students make extensive use of online resources in lower-division mathematics classes. This explanatory mixed method study (Creswell & Plano Clark, 2011) explores how and why these students are using online resources for their self-directed learning in mathematics courses. Using a survey of 108 students from 13 2-year and 4-year colleges and 26 follow-up interviews, we identify the most commonly used online resources, self-reported description of how these resources are used, and students’ perceptions of best practices as well as their concerns about maladaptive practices. We use an Activity Theory framework (Engeström, 1999) that directs our attention to the role that tools play in students’ development over time and how these tools are leveraged to achieve learning goals. Our findings reveal undergraduate students as thoughtful and deliberate learners who make extensive use of online resources to help navigate the challenges of higher education.
2020
Boston, Massachusetts
Combinatorial proof is an important topic both for combinatorics education and proof education researchers, but relatively little has been studied about the teaching and learning of combinatorial proof. In this paper, we focus on one specific phenomenon that emerged during interviews with mathematicians and students who were experienced provers. In particular, participants used a wide variety of cognitive models to interpret multiplication by a constant when reasoning about binomial identities, some of which seemed to be more (or less) effective in helping produce a combinatorial proof. We present these cognitive models and describe episodes that illustrate implications of these cognitive models for our participants’ work. Our findings both inform research on combinatorial proof and highlight the importance of understanding subtleties of the familiar operation of multiplication.
2020
Boston, Massachusetts
In this paper I discuss the process of creating a closed-form multiple-choice assessment of students’ ability to validate mathematical arguments at the introduction to proof (ITP) level. This process involved: (1) creating and validating a framework of common validity issues (CVI) in proof writing as a basis for assessment creation through a mathematician survey (𝑛 = 228) and two focus groups (𝑛 = 4 & 𝑛 = 7); (2) creating and piloting an open version of the assessment as a means to create distractors for the closed assessment; (3) creating, piloting (𝑛 = 187) and analyzing the results from the closed form assessment; and (4) conducting interviews with student participants after the pilot to determine the process that students took during the pilot. The results of the processes offer an assessment that, with some refinement, can measure students’ ability to validate mathematical arguments from several perspectives in the ITP setting.
2020
Boston, Massachusetts
This paper reports on several aspects of a larger study aiming to characterize the engagement of students in a Precalculus class at a four-year public university. In line with policy suggestions that advocate for the development of flexibility in mathematics problem solving, we conducted a teaching experiment which utilized a set of "multiple solutions activities" to expose students to alternative solution methods while providing opportunities to examine and critique the reasoning of others. In addition, the instructor and teaching assistant of the course attempted to use these activities to explicitly negotiate productive norms and practices. Nevertheless, several detrimental norms emerged in the class, as the effectiveness of the activities was hindered by several propensity factors, particularly students' prior knowledge and self-regulation.
2020
Boston, Massachusetts
MathChavrusa is a novel application of Talmudic study techniques into the context of mathematics education, emphasizing long-term, partnered text study and problem solving. This study implemented MathChavrusa in semester long graduate mathematics courses, collecting data from both instructor and student perspectives on the effectiveness of MathChavrusa in facilitating student understanding of course material and engagement in communicating mathematics. Results indicated that students largely had a positive impression of MathChavrusa, highlighting its impact on student engagement, creating a conducive environment for asking peer-peer and peer-instructor questions, and was a factor in better understanding course materials. This study model has the potential to enhance student engagement in mathematics classrooms and could be adopted more broadly.
2020
Boston, Massachusetts
Prior research has shown that students tend to reason in terms of the informal components of their concept images rather than rely on formal concept definitions when working with topics in mathematics. However, students’ concept images may be incomplete or inaccurate, or students’ evoked concept images may exclude important features of concepts or classes of examples. This report describes two students’ evoked concept images of concepts in general topology as revealed through their completion of proof tasks as well as some of the effects those images had on their success in proving. We also discuss the impact of the students’ exploration of their example spaces on their concept images and on their proof writing.
2020
Boston, Massachusetts
Previous literature has yet to document whether students do not read their mathematics textbooks because they choose not to or because students are unable to read them effectively. This report is a case study investigating the relationship between a student’s beliefs about textbooks to learn mathematics and their readings of mathematical text. We find that students may choose not to utilize textbooks as part of their learning despite being effective mathematical readers. Future research should investigate the relationship between what students think it means to understand mathematics and their beliefs about learning through interactions with mathematics textbooks.
2020
Boston, Massachusetts
Inquiry-Based Learning (IBL) teaching practices have been hard to characterize in mathematics education research, as mathematics instructors interpret and engage with these practices in different ways. Using systemic functional linguistics (SFL), specifically the appraisal framework (Martin & White, 2005), we analyzed responses to the question “what is IBL” given by 12 mathematics instructors who were teaching with IBL. Our analysis was guided by the following question: What do the language choices of a group of undergraduate mathematics IBL instructors reveal about their perceptions of IBL? We identified how the instructors opened up, closed down, and changed the strength and focus of their messages through their grammatical and lexical choices. We found that instructors softened their language when defining IBL but used sharper and intensified language with less room for negotiation when talking about teacher and student roles in IBL classrooms.
2020
Boston, Massachusetts
In this study, we provide an analysis of opportunities for undergraduate students to engage in the skill of reasoning-and-proving in a precalculus textbook in a medium-sized university in North Eastern USA. To investigate these opportunities, we focused our analysis on the frequency and nature of reasoning-and-proving opportunities available in both the exercises and narrative section of the book. Findings indicate that there were more reasoning-and-proving opportunities in the narrative section of the book than in the student exercises section. Furthermore, the opportunities in the student exercises section were predominantly of the particular nature while those in the narrative section were general. Implications of these findings are discussed.
2020
Boston, Massachusetts
This study reports results of an investigation into the meanings a precalculus student constructs as she views an animation independently and describes her image of the problem context as it influences her ability to construct meanings in a novel problem context. Using an exploratory teaching interview, the student’s initial meanings and image of the problem context shifted once the animation was utilized as a didactic object. An analysis of her meanings and prior research are used to modify a hypothetical learning trajectory for supporting students in their emergence of productive images of novel problem contexts to construct meaningful and appropriate formulas and graphs. The results and the construction of a hypothetical learning trajectory of this study can potentially be useful for a teachers’ awareness of students’ constructed meanings while designing or selecting mathematical tasks that incorporate applets and animations in her instruction.
2020
Boston, Massachusetts
This study reports psychometric evidence for using the high school version of the revised Self-Efficacy to Teach Statistics (SETS-HS) instrument in a retrospective pre-post format with for pre-service mathematics teachers. Analyses at the subscale level indicate adequate internal consistency for “before” and “now” ratings and a high correlation between subscale scores across both ratings. Analyses of individual items’ item means, item-subscale total correlations, and the frequency of item mentions in open-ended items are summarized. The retrospective pre-post format yielded SETS-HS scores with some adequate psychometric properties, suggesting a single administration instead of two separate administrations does not degrade the measurements.
2020
Boston, Massachusetts
Proving can be a process for communicating mathematics and people communicating need to be familiar with the language of mathematics. This study focuses on the prompts used for proving tasks not only because the wording of these prompts may impact students’ modes of argumentation, but also because students might bring their preconceptions about these prompts from previous mathematical and non– mathematical experiences. The current study examined how Calculus I students interpret prompts, such as “prove” and “show.” Data was from 131 survey responses and three interviews reveal that these students possibly interpret different prompts in different ways, which can create a gap between students’ interpretations and instructors’ or researchers’ intentions. This suggests that instructors and researchers may need to be more deliberate in choosing such prompts.
2020
Boston, Massachusetts
We report results from administering a paper-and-pencil survey to 56 future middle grades and secondary grades mathematics teachers enrolled in teacher preparation programs at two large public universities in the Southeast. We intentionally designed one task on the survey so that it could be used with the variable-parts perspective on proportional relationships. The task asked teachers to show relationships among quantities described in the text by drawing a picture of the situation, writing an equation, and writing a few sentences to explain their equation. Analysis of the responses led to three results: (a) most future teachers represented relationships between parts and wholes appropriately, either by comparing parts to parts or by combining parts to make wholes, (b) future teachers were successful using division as often as multiplication when writing linear equations that combined two different units, and (c) when future teachers made errors using multiplication, reversal errors were the most common.
2020
Boston, Massachusetts
Pre-service mathematics teachers need to have an advanced perspective of geometry concepts and be able to extend and generalize the geometry content they will teach. This study documents the continued refinement of a geometry lesson using computer programming and GeoGebra to iteratively teach generalization over the concept of interior and exterior angles in polygons and, ultimately, helps to refine a previously developed genetic decomposition for continued improvement of this and future lessons.
2020
Boston, Massachusetts
Students learn more deeply when conceptual understanding is at the forefront and connections are made between topics. With this in mind, we have created a hypothetical learning trajectory (HLT) for the chain rule, related rates, and implicit differentiation to teach them in a conceptual, connected way. In a previous paper we outlined the creation of the HLT based on nested multivariation (NM). In this second paper we describe a small-scale teaching experiment done to test the HLT. Our results suggest NM was an appropriate construct to base the HLT on, and we report on these students’ developing understandings. Based on the results, we made final adjustments to the HLT in preparation for a full-scale classroom teaching experiment.
2020
Boston, Massachusetts
Drop-in tutoring has been shown to positively impact grades. However, little is known about what occurs in drop-in tutoring. This study sought to fill this gap by answering the following research questions: 1) How do tutors interact with students? 2) What factors influence tutors’ decisions regarding how to navigate interactions? 3) How do these factors impact tutors’ interactions? This study provides a descriptive case study of one tutor’s practices in the context of drop-in undergraduate Calculus I tutoring. Natural tutor session recordings and stimulated recall interviews were collected. Multiple lenses, including teacher decision-making, social and socio-mathematical norms, responsive teaching, and cognitive apprenticeship, were utilized during analysis. Results found tutor-student interaction patterns are more complex than previously reported in the literature. In addition, results suggest tutors are capable of changing their practices within the short time span of a semester and further, tutors are capable of enacting “best practice” teaching strategies.
2020
Boston, Massachusetts
Research shows that students in mathematics can benefit from participating in self- and peer-evaluation. However, many students lack the self-confidence in their ability to grade their peers’ work and often provide feedback that is unhelpful. This study investigates the criteria used by a student to evaluate responses to questions on an assignment pertaining to comprehension of a proof by contradiction. Two interviews were conducted with the student more than one year apart in which she evaluated her own responses and responses written by an instructor to the questions in the assignment. Through the lens of achievement goals, results provided show a shift in the way she used her criteria to provide evaluations of these responses. Further, analysis indicates students may benefit from regular activities and practice in self-/peer- evaluation of proofs without assigning a grade to help their reflection and sense-making abilities.
2020
Boston, Massachusetts
This study focuses on a student’s thinking about a constant rate of change in the context of a conceptual approach of calculus (DIRACC). DIRACC calculus defines a constant rate of change as variations measured in two covarying quantities being proportional to each other. This paper also presents a theoretical framework that characterizes students thinking about the constant rate of change in relation to variation and covariation.
2020
Boston, Massachusetts
When talking about mathematics, teachers and learners actively use hand gestures to support their speech as well as to describe ideas that are not expressed verbally. In this study, I investigate the gestures that were utilized by an instructor and his students during a teaching episode on proof by mathematical induction. Alibali and Nathan’s (2012) typology of gestures are employed to code the observed gestures. The study reveals that the use of gestures plays an integral role in teaching and learning induction. I show that pointing gestures helped to reduce ambiguity in classroom discussion, representational gestures were useful in describing specific subcomponents of induction, and, finally, metaphoric gestures were independently introduced by a teacher and students to describe the nature of proof by mathematical induction.
2020
Boston, Massachusetts
Balacheff (2008) offers the notion of epistemology of proof to account for researchers’ views on what a mathematical proof is from a teaching-learning perspective. This study argues that an awareness of mathematicians’ epistemologies of proof might provide insight into their rationales and actions when teaching university students how to prove. This argument is illustrated by the case of one mathematician who provided comments and marks on proofs submitted by her graduate students as part of her course instruction of topology. A commognitive framework is mobilized to discern the mathematician’s epistemology of proof.
2020
Boston, Massachusetts
This study builds on existing research detailing calculus students’ algebraic errors (“gaps”) by connecting data from a standardized, proprietary placement instrument. Results suggest skills that are rarely versus commonly demonstrated prior to calculus. We also report on success and failure for students in the STEM-track Calculus 1 based on the different skills.
2020
Boston, Massachusetts
This paper shares results of a discourse analysis of an interview with a female student who was working as an undergraduate mathematics tutor and learning assistant. The student is shown to have developed a mathematics identity made up of a strong identity as a doer of mathematics and a weak sense of belonging within her mathematics community. I use robust mathematics identity to refer to a mathematics identity that includes both a strong identity as a doer of mathematics and a sense of belonging in mathematics. A fragile mathematics identity is one in which one or the other of these constructs is not present. This interpretation of these terms is closely aligned with, but slightly different from, the way these terms are used in other literature in the field. This student is shown to have a fragile, rather than robust, mathematics identity.
2020
Boston, Massachusetts
In an attempt to address the problem of discontinuity between school mathematics and collegiate mathematics, the current study aimed at characterizing how university students’ previous knowledge constructed in school mathematics can be reorganized in their learning of collegiate mathematics. To this end, a construct called transformative transition and its accompanying categorical framework were developed and explained with student data. A transformative transition involves a qualitative leap in existing understandings as an individual encounters a new construct and integrates it into his/her cognitive system. Its four categories—extending, deepening, unifying, and strengthening—delineate different ways in which that qualitative leap might take place. Results from the analysis of interviews with six mathematics-intensive majors revealed that the context in which the participants could actively revisit, observe, reflect on, and interrelate their existing understandings at a higher level and from a different angle helped them to reorganize their school mathematics understandings.
2020
Boston, Massachusetts
This study examined how eight students in an introduction to proof (ITP) course viewed a “cheating scandal” where their peers submitted homework containing solutions found on the web. Drawing on their weekly log entries, the analysis focuses on the students’ reasoning about the difference between acceptable and unacceptable use of internet resources in learning mathematics. One pattern was that students’ view of the relationship between beliefs about mathematics and the work of learning mathematics grounded their views of “cheating.” Specifically, some felt that an implicit didactical contract required that model solutions should be available when one learned new material. The case raises the general issue of the relationship between the process of learning mathematics and the appropriate use of external resources. It suggests that instructors may need to re-examine the role of homework, especially its assessment, in their courses, so that productive struggle is valued, not avoided.
2020
Boston, Massachusetts
Secondary mathematics curricula predominantly present linear programming by introducing the corner point principle and formulating steps for a solution. The teaching and learning of this topic often overlook justification for the principle and procedures. Drawing on six clinical interviews with an in-service teacher, we illustrate how her covariational reasoning supported conceiving quantities’ multi-variation entailed by an objective function and its constraints (a system of inequalities). We build upon this teacher’s understandings to propose a conceptual analysis for optimizing two-variable objective functions in the context of linear programming.
2020
Boston, Massachusetts
We investigate how students make sense of irrational exponents. The data is comprised of 32 interviews with university students, which revolved around a task designed to examine students sensemaking processes involved in the understanding of the concept. Both the task design and data analysis relied on the concept of sensemaking trajectories, blending the notions of sensemaking and (hypothetical/actual) learning trajectories. The findings focus on three kinds of reasoning utilized in the participants’ sensemaking trajectories while working with irrational exponents: even/odd numbers and functions; range of exponent values; and exponentiation as repeated multiplication. These findings reveal students’ conceptual development of irrational exponents, and could in turn be used for the refinement of tasks aimed at promoting students’ comprehension of the topic.
2020
Boston, Massachusetts
One of the ways in which university math departments across the country are making efforts to improve their introductory math courses is by implementing or increasing the level of course coordination for their Precalculus to Calculus 2 sequence. This not only entails creating uniform course elements across different sections but also includes efforts to build a community among the instructors of the course. While many coordinators have the common goal of improving student success, we explore what guides their actions to see this accomplished, what we refer to as their orientation toward coordination. The orientation of a coordinator encompasses their beliefs, values and knowledge of mathematics and teaching. In this proposal we introduce and elaborate on two orientations toward coordination that arose from interviews with course coordinators from a variety of institutions across the country. We also discuss the importance of both orientations as they relate to drivers of change.
2020
Boston, Massachusetts
We present results of a grounded analysis of individual interviews in which students play Vector Unknown - a video game designed to support students who are taking their first semester of linear algebra. We categorized strategies students employed while playing the game. These strategies range from less-anticipatory button-pushing to more sophisticated strategies based on approximating solutions and choosing vectors based on their direction. We also found that students focus on numeric and geometric aspects of the game interface, which provides additional insight into their strategies. These results have informed revisions to the game and also inform our team’s plans for incorporating the game into classroom instruction.
2020
Boston, Massachusetts
Students’ epistemologies (beliefs about mathematical knowledge) play a significant role in how they learn mathematics. Much prior work in this area describes student epistemologies as unitary, where evidence of a particular epistemology is taken to represent that student’s singular epistemology of mathematics. Most argue that these epistemologies stem from their experiences across their mathematics classes. This study set out to examine students’ epistemologies in introductory calculus and how those epistemologies influence and reflect students’ experiences in mathematics courses. To a significant extent, it confirms results from prior studies. However, through one-on-one it finds more variability in students’ epistemologies than most of the literature has described. That variability is significant for helping educators think about how to help students learn mathematics in general, and calculus in particular.
2020
Boston, Massachusetts
Evaluating the Quality of Instruction in Post-secondary Mathematics (EQIPM) is a video coding instrument that provides indicators of the quality of instruction in community college algebra. Following an extensive revision, and the coding of a set of 84 videos from 40 instructors, we report the results of a confirmatory factor analysis, that suggests that the instrument captures three distinct hypothesized dimensions of quality of instruction in community college algebra classes. Due to the nature of the instruction observed, the items in the instrument needed to be treated as categorical and some had to be removed from the analysis. We use the data to provide a first glimpse of the quality of algebra instruction that we found in this data set.
2020
Boston, Massachusetts
Typically, departments and universities evaluate and provide feedback to college mathematics instructors on their teaching by conducting classroom observations. When these observations are guided by an observation protocol, the protocol provides a particular lens that focuses the observer’s attention on certain aspects of instruction. To understand the way in which mathematics departments are currently evaluating teaching, we investigated the structure, focus, and alignment with inquiry-based mathematics education practices of items on 25 observation protocols. The results suggest that protocols mainly include closed-response items that are evaluative, focus more on the instructor than the students or the content, and rarely attend to inquiry-based mathematics education practices. These results have implications for both those who develop observation protocols and those who use observation protocols as a measure of teaching practices. In particular, our study highlights the need for careful consideration to be taken when observations are used for performance measures.
2020
Boston, Massachusetts
This paper extends work in the area of quantitative reasoning at the undergraduate level, in addition to proposing a conceptual framework for interpreting partial derivatives in different contexts. Task-based interviews were used to examine third-semester calculus students’ reasoning about partial derivatives in two tasks, one situated in a mathematics context (Alajmi, 2012) and the other in a diet and exercise context. Findings of this study indicate that interpreting partial derivatives, especially in the latter context, was problematic for a majority of the students. Overall, findings of this study suggest that student difficulties with interpreting partial derivatives in real-world contexts are similar to student difficulties with interpreting ordinary derivatives. Implications for calculus instruction are included.
2020
Boston, Massachusetts
Quadratic relationships are an important topic in algebra through calculus. In this report, we describe how a task designed using Realistic Mathematics Education (RME) principles supported students in developing meanings for quadratic relationships via their covariational reasoning. After describing the task design and sequence, we present student activity highlighting shifts in their meanings before, during, and after a six-week teaching experiment. We highlight the role of interactions amongst students, task, and the teacher in supporting such shifts. We conclude with a discussion and areas of future research.
2020
Boston, Massachusetts
This paper describes ways of thinking students employ when they choose to use calculations or produce algebraic expressions to respond to mathematical tasks and their expectations regarding the meanings of what they produce. My findings suggest that students’ reasoning in symbolization activity is often guided by perceptual features of tasks, such as the numbers explicitly given in prompts and key words students identify. I describe the construct “emergent symbolization” as a potentially productive way of thinking in symbolization activity based on synthesizing prior work and the results of this study. I close by making connections between similar work in analyzing students’ ways of thinking about graphs and discussing how my work contributes to the common instructional goal of promoting connections between representations.
2020
Boston, Massachusetts
This paper examines the mechanism behind students’ guided reinvention process of a formal definition of the limit of a sequence by looking at how students’ shared understanding develops and what discursive rules enable such development using the commognitive approach focusing on the relationship between routines (discursive rules) and changes in endorsed narratives (shared understanding). A Calculus II instructor conducted a teaching experiment in which 11 students reinvented a formal definition of the limit starting from their initial statement about the sequence convergence, mainly including dynamic language such as “approaching.” In our analysis, we identified the routines about the application of the definition in examples and non-examples, and the consistency between written and illustrated form of their definition, and the readers of the definition, and how those routines contributed to the changes in students’ endorsed narratives about sequence convergence during the guided reinvention.
2020
Boston, Massachusetts
Graphs of real-valued functions figure prominently in the study of calculus. While the use of graphs in instruction has shown promise for supporting student thinking, previous research has also shown that students may interpret graphs in ways that differ from an instructor’s intention. This study aimed to investigate how students interpret expressions from calculus statements in the graphical register. To this end, I conducted 150-minute clinical interviews with 13 undergraduate mathematics students who had completed a calculus course. In the interviews, students evaluated six calculus statements for various real-valued functions depicted in graphs in the Cartesian coordinate system. I describe the characteristics of four distinct interpretations of expressions from these statements in the graphical register that students used in this study, which I refer to as (1) nominal, (2) ordinal, (3) cardinal, and (4) magnitude. I discuss some implications of these findings for teaching and directions for future research.
2020
Boston, Massachusetts
While learning is often characterized as a process of abstractions from robust understandings, students’ reasoning in advanced contexts can be a more dynamic process of reflection across multiple layers of complex cognitive structures. To explore the ways that real analysis students’ understandings of abstract mathematical concepts evolve in the context of multi-faceted proofs, we observed students’ attempts to prove the Arzel`a-Ascoli Theorem. We describe the understandings from which students built to engage in more complex reasoning and demonstrate their folding back (Pirie & Kieren, 1994) to facilitate the co-evolution of their foundational and advanced understandings.
2020
Boston, Massachusetts
This case study utilizes self-determination theory (Deci & Ryan, 2000) to explore the motivational development of three students who participated in problem posing as part of an introduction to proofs course. The experience of these students illustrates several ways in which problem posing can create new more integrated habits of motivational regulation. However, the differences of these cases highlight how students may benefit from experiencing problem posing in a way that makes explicit the purposes of student problem posing in relation to other educational goals. This research underscores the motivational benefits of making the connections between problem posing, problem solving, learning, and self-regulation more explicit earlier in students’ mathematical education.
2020
Boston, Massachusetts
Group isomorphism and homomorphism are topics central to abstract algebra, yet research on instructors’ views of these concepts is limited. This study examines two instructors’ views of group isomorphism and homomorphism and how they drew on notions of sameness to describe them. Differences between instructors’ descriptions in interviews and instruction are explored.
2020
Boston, Massachusetts
Students’ characterizations of their errors have the potential to direct their future efforts in mathematics, both in and out of the classroom. This study investigates the errors made on one exam by three college algebra students who identified “simple mistakes” as the primary reason they were not satisfied with their test grade. We developed codes for the types of errors made and classified each as “simple” or “not simple.” We then calculated the percentage of points lost due to simple mistakes for each of the three students. Results showed that for each student, approximately 20-40 percent of the errors made on this test could be classified as simple mistakes. We present hypotheses connecting the students’ frustration to these results and provide directions for future research that may test these hypotheses and illuminate differences between student and instructor definitions of simple mistakes.
2020
Boston, Massachusetts
The transition to proof is a known point of struggle for many undergraduate mathematics majors. This work examines this difficult transition point from an affective lens, specifically students’ affective pathways while working on proof construction tasks. A series of four semi-structured interviews were conducted with 11 undergraduate students enrolled in a transition to proof course, and affective pathways were studied using non-traditional instruments: an emotion word selection task and an emotion graph. Open coding and constant comparative methods were used to identify common affective pathways within the emotion words and graphs. These pathways reveal common student experiences while working on proofs, which may be used to fine tune external learning conditions to positively influence engagement and experience.
2020
Boston, Massachusetts
The purpose of this study is to investigate students’ engagement with and utilization of the proof feedback from their professor, and their feedback preferences. We interviewed seven participants about their general experience with feedback on proofs. We then continued interviewing four of these students periodically about the feedback they received on specific selected proofs to see how they interpreted the feedback they received and how they utilize the feedback when asked to revise their proof production. The results showed that the majority of participants had a strong preference towards explicit feedback and, in contrast, that professors provided primarily implicit feedback. Furthermore, we found that students did not read professors feedback when they received satisfactory grades on their proofs, they usually only utilized the feedback when working subsequent assignments or studying for exams, and they did not use the Professors’ feedback to rewrite their proof production.
2020
Boston, Massachusetts
We explored how 12 quantum mechanics students from two universities discussed basis and change of basis as they performed probability tasks, one of which required a change of basis. We found that students’ utterances referred to a person, a calculation, or a vector being in a basis, or a vector being written in a basis. Students discussed change of basis as changing the form of a vector, writing the vector in another form, making a vector become a new vector, and switching bases. We performed a discourse analysis on the situated meanings of students’ phrases.
2020
Boston, Massachusetts
In this study of inquiry-oriented instruction (IOI), we explore the relationship between beliefs, professional obligations, and inquiry-oriented practice among college mathematics instructors. Professional obligations consist of the responsibilities that instructors have towards various stakeholders, including the institution, the individual students, mathematics as a discipline, and society (Herbst & Chazan, 2012). Past studies have reported inconsistencies between beliefs and practice; professional obligations may help explain why instructors cannot always realize IOI practices in the classroom. We operationalize these constructs in a set of surveys and use structural equation modeling to explore the hypothesized relationships.
2020
Boston, Massachusetts
Felix initially used a metaphor in considering translations between algebraic representations of systems during an individual clinical interview. I investigated the mathematical underpinnings for his metaphor by asking him to perform translations in the vector space register. That investigation precipitated a discussion where Felix developed mathematical understanding for a translation between the vector space register and the linear systems register. My analysis is based on the Theory of Quantitative Systems which I developed as a result of my study of Duval’s Theory of Semiotic Representation Registers (1999, 2006, 2017).
2020
Boston, Massachusetts
To understand linear algebra concepts, one needs to be familiar in many different modes of thinking. Mathematicians often move between these modes of thinking fluently and expect that students will pick up the main ideas along the way. Yet, a majority of students do not have the cognitive framework to perform the move that is so natural to mathematicians. Employing Tall’s (2013) three-world model, in this research a set of linear algebra tasks were given to students in order to encourage them to move between Tall’s embodied, symbolic and formal worlds of mathematical thinking. Our working hypothesize is that by creating opportunities to move between the worlds, students may be exposed to multiple modes of thinking which results in richer conceptual understanding. The results of a survey revealed that a majority of students preferred the symbolic world. Their reasoning for their choices will be discussed.
2020
Boston, Massachusetts
We present the results of a teaching experiment designed to foster a pre-service secondary teacher’s construction of a scheme for constant rate of change through engendering reflecting and reflected abstractions. Although the research participant developed a productive conception of rate of change as an interiorized ratio, images of chunky continuous covariation imposed an obstacle to her ability to reason efficiently across a variety of contexts. The participant’s rate of change scheme was grounded in reflected abstraction, which enabled her to become consciously aware of its essential aspects and to appreciate its applicability across a variety of contexts.
2020
Boston, Massachusetts
In this study, we focus on a student’s meanings for lines and points in the context of graphing covarying quantities. Specifically, we illustrate a student conceiving a line as representing a direction of movement of a dot on a coordinate plane. Consequently, the student did not conceive a dot moving in the coordinate plane as leaving a trace of infinitely many points; similarly, points on a line did not exist until they were physically and visually plotted. We conclude that the student’s meanings for lines and points had a significant impact on his graphing activities, in particular, on his construction of emergent shape thinking.
2020
Boston, Massachusetts
As attention grows towards the disparities between majority groups and underrepresented minorities within undergraduate STEM education, there is a need for understanding where different university stakeholders stand on the subject of increasing diversity. This paper aims to juxtapose categorizations of stakeholder motivations and dominant and critical perspectives to propose a framework for analyzing stakeholder attitudes and informing increasingly productive conversations on eliminating inequity in STEM fields.
2020
Boston, Massachusetts
Fostering students’ mathematical creativity necessitates certain instructional actions - one of which is designing and implementing tasks that foster creativity. Drawing on the literature on mathematical creativity, we describe existing research-based features of tasks for eliciting student creativity, or creativity-based tasks, and provide suggestions for implementation of such tasks. Based on these features, we analyzed two instructors’ first experiences designing and implementing creativity-based tasks in Calculus I. Both instructors’ frequent use of the multiple-solutions feature suggests that this feature could be an entry-point for designing and implementing creativity-based tasks for other instructors seeking to foster creativity.
2020
Boston, Massachusetts
Interpersonal relationships are central to the teaching and learning of mathematics. One way that teachers relate to their students is by experiencing empathy for them. In this study, I explore the phenomenon of pedagogical empathy, which is defined as empathy that influences teaching practices. Specifically, I examine how graduate student instructors (GSIs) conceptualize empathy as it relates to the teaching and learning of mathematics and identify factors that influence this pedagogical empathy. The findings presented in this paper emerged from interviews with 11 GSIs in which they answered questions designed to elicit their thoughts about empathy. The participants provided complex and diverse notions of how empathy relates to teaching mathematics. Conclusions drawn from this paper may help inform professional development efforts targeted at novice and experienced instructors at the post-secondary level.
2020
Boston, Massachusetts
This paper reports a qualitative study of how small group problem solving was enacted differently across sections of a multi-section undergraduate introduction to proof course. Common course materials, common guidelines for instruction, and weekly instructor meetings led by a faculty course coordinator supported similar instruction across sections, including an emphasis on in-class group work. But within that shared structure, classroom observations revealed important differences in how group work was introduced, organized, and managed. Our results focus on differences in the time allotted to group work, the rationale for group work, the selection and organization of groups, and aspects of student activity and participation. We suggest that these differences shaped different opportunities to learn proof writing in small groups. These results have implications for the design and teaching of collegiate mathematics courses where group work is a regular element of classroom work.
2020
Boston, Massachusetts
Quadratic and exponential relationships are important topics in school mathematics. However, there is limited research examining students’ understandings of these relationships. In this paper, we present data from a semester long teaching experiment with a pre-service teacher in which we examined ways in which she could leverage covariational reasoning to differentiate between quadratic and exponential change. Whereas in the pre-interview, the student did not have meanings that supported her in differentiating between the relationships, her experiences in the teaching experiment supported her in developing more robust meanings. By the end of the teaching experiment, the student could differentiate between quadratic and exponential change.
2020
Boston, Massachusetts
Research literature suggests that students struggle in gaining fluency with mathematical language, and especially with statements with multiple quantifiers. I conducted a design study rooted in Realistic Mathematics Education in order to support students in developing such fluency. Drawing on the emergent model heuristic, I present a pair of students’ reinvention of relationships between variables in statements with multiple quantifiers. This work is a case study for how students might engage in defining in order to learn how and in which order to use quantified variables. Relationships between variables first emerged as a model-of a pair of students’ activity as they defined different concepts that are best defined using multiple quantified variables. Additionally, these relationships became more explicit to the students as they reflected on their definitions.
2020
Boston, Massachusetts
The purpose of this paper is to highlight the reasonings students use to move between representations of multivariable functions. We identify five ways students use of aligning graphical representations of functions, or surfaces, with contour maps. Subsequent tasks highlight the affordances one particular reasoning, called Contour Spacing Alignment, offers for students to further develop their reasoning. We analyze student reasoning highlighting the transition between model-of and model-for this reasoning.
2020
Boston, Massachusetts
Students encounter advanced mathematical concepts in both mathematics classes and physics classes. What meanings do they develop about the concepts across the various contexts? Our research project investigates students' meanings for eigentheory in quantum mechanics and how their language for eigentheory compares and contrasts across mathematics and quantum physics contexts. We present students' interpretations of a canonical mathematical 2x2 eigenequation, a spin-½ operator eigenequation, and a spin-½ operator equation in which the operation "flips" the spin state. In individual, semi-structured interviews, 9 quantum mechanics students were asked to explain what the first two equations meant to them and then to compare and contrast how they conceptualize eigentheory in the two contexts. They were then asked to discuss the third equation. Using discourse analysis, results characterize students' nuanced imagery for the equations and highlight instances of synergistic and potentially incompatible interpretations.
2020
Boston, Massachusetts
In this small-scaled classroom teaching experiment we investigated the nature of six preservice secondary mathematics teachers’ in-the-moment engagement and collective mathematical activity as they worked in pairs. Students participated in five 1-hour sessions focused on concepts of logarithms and relationships between linear and multiplicative change. We analyzed pair- and whole-group argumentation using Toulmin models (1958\2003) to understand collective mathematical activity and individuals’ participation in argumentation. Self-reported data on student engagement were collected at two random times per session per student, which were further explicated through recall interviews. Results suggest active involvement in mathematical argumentation is not sufficient for high engagement. Instead, contributing to mathematical ideas functioning as-if-shared which also prompt deeper understanding of ongoing work align with states of relatively high engagement.
2020
Boston, Massachusetts
Acknowledging the significant contribution of mathematicians to the mathematical education of teachers, we explore the views of mathematicians on an envisioned Calculus course for prospective teachers. We analyzed the semi-structured interviews with 24 mathematicians, using the EDW (Essence-Doing-Worth) framework (Hoffmann & Even, 2018, 2019); and subsequently, we adapted the framework by extending and refining the existing themes. The findings of our study indicate that the mathematicians believe the primary purpose of a Calculus course for teachers is to communicate the nature of mathematics as a discipline. By providing a variety of examples that could shape and expand the teachers’ understanding of mathematics, the majority of the mathematicians participated in the study emphasized the value of mathematical investigation in an envisioned Calculus course for teachers, as well as connections within and beyond the subject.
2020
Boston, Massachusetts
College and department administrators take undergraduate student complaints about Graduate Student Instructors (GSIs) seriously. However, little research has been done to examine the nature of undergraduate student complaints across multiple mathematics departments from the lens of student-centered instruction. In this study, we compared formal (i.e. documented in writing by the student) undergraduate mathematics student complaints about GSIs at two universities over five years. Complaints were analyzed by coding the contextualized concerns described in the complaints using the Mathematical Association of America’s Instructional Practices Guide to align complaints with topics discussed as best-teaching practices. Results demonstrated that concerns about classroom and assessment practices were the most prevalent. Concerns about classroom practices were slightly more abundant and more pervasive throughout the semester than concerns about assessment practices. Additionally, an outside-of-class issue undergraduate students raised was regarding the effectiveness of GSIs communication via emails.
2020
Boston, Massachusetts
The purpose of this study is to examine students’ meanings for the derivative at a point. While students may associate rate of change with derivative, this does not mean that the meaning they have for derivative is productive. This study explores students’ responses to a typical calculus 1 problem that uses derivative to determine a linear approximation. A framework is provided for describing and analyzing students’ meanings for the derivative at a point by discussing students’ usage of time and meaning for change in the context of a fish growing over time.
2020
Boston, Massachusetts
Students can learn more deeply when conceptual understanding is at the forefront and connections are made between topics. We hypothesize that such understanding and connections can be achieved for the chain rule, implicit differentiation, and related rates through the construct of nested multivariation (NM). In this first paper, we describe the process of creating a hypothetical learning trajectory (HLT) rooted in NM for this sequence of topics. This theoretical paper contains our conceptual analysis, literature review, and construction of the HLT.
2020
Boston, Massachusetts
This paper reports findings from task-based interviews with 6 mathematicians, conducted to document mathematicians’ in situ application of their orientations towards proof by contradiction; that is, their reflective dispositions towards proof by contradiction. However, analyses suggest that when engaging in situated analyses mathematicians’ behaviors are better characterized as proof repertoires: sets of action-based dispositions tacitly codified into relationally-determined proving routines and best thought of as replicable agent-tool interactions. This theoretical report introduces these constructs (i.e., orientations and repertoires), and illustrates the utility of the latter when interpreting mathematician’s ways of reasoning about proofs by contradiction in situ.
2020
Boston, Massachusetts
I devised the Theory of Quantitative Systems (Sipes, 2019) as a lens for considering the complexity of comprehending linear algebra. The theory resulted from a study of Duval’s (1999, 2006, 2017) Theory of Semiotic Representation Registers. With my theory I consider strictly algebraic contexts for systems of linear equations. This report discusses some of the theory’s details which may provide insights into challenges encountered in linear algebra courses. Appendix A is a one-page presentation of how I employed Duval’s theory in creating my own
2020
Boston, Massachusetts
In this discussion, we frame an argument to no longer invoke the Mathematical Knowledge for Teaching (MKT) framework (Ball, Thames, & Phelps, 2008) at the post-secondary level of mathematics education research, since it is theoretically based in the work of elementary school teachers. After reviewing the MKT components and related research regarding the extension and modification of the MKT framework to higher mathematics education, we propose a recasting of Shulman’s (1986, 1987) initial formulation of content knowledge for teaching as a route for developing a unifying framework for mathematics education. This report uses Precalculus adjunct instructors’ interview data to propose a coding framework by applying Shulman’s principles and categories of content knowledge.
2020
Boston, Massachusetts
Textbooks are one of the most important tools teachers and students use in any course. Thus, textbooks can greatly influence what occurs in classrooms. Textbook analysis provides insight into the rolls and effects of textbooks have on the potentially implemented curriculum for any course. This paper proposes a new framework that builds from several existing ones while introducing a focus on analyzing the authors’ conveyed meanings by drawing upon Thompson’s theory of meanings. While developed to address a need for textbook analysis in introductory statistics, the proposed framework can work well in mathematics and at different levels (i.e., not just introductory courses). Further, the proposed framework is a starting point for developing a more robust framework.
2020
Boston, Massachusetts
Until recently, the variable-parts perspective on proportional relationships had been overlooked in mathematics education research. We argue that the variable-parts perspective is implicit in contexts of geometric similarity, including trigonometry and the notion of slope, and that it is well-suited to situations involving varying increments of change, as in calculus. We explain how a variable-parts approach to these topics is related to approaches and findings of research on trigonometry, rate of change, and the Fundamental Theorem of Calculus. Our arguments are theoretical, but lead to the empirically testable hypothesis that an explicit focus on the variable-parts perspective could be productive for learning geometric similarity, trigonometry, slope, and calculus.
2020
Boston, Massachusetts
In this paper, I propose a theorization of mental processes involved in teachers’ learning of students’ mathematics in the context of student-teacher interaction. Specifically, I combine the Piagetian scheme theory with the notions of first- and second-order modeling to characterize types of mathematical learning that may occur when a teacher decenters. I also discuss the affordances and methodological considerations in light of the proposed framework.
2020
Boston, Massachusetts
Researchers are producing a growing number of studies that illustrate the importance of quantitative and covariational reasoning for students’ mathematical development. These researchers’ contributions often are in the context of learning of specific topics or developing particular reasoning processes. In both contexts, researchers are detailed in their descriptions of the intended topics or reasoning processes. There is, however, a lack of specificity relative to generalized criteria for the construction of a concept. We address this lack of specificity by introducing the construct of an abstracted quantitative structure. We discuss the construct, ideas informing its development and criteria, and empirical examples of student actions that illustrate its use. We also discuss potential implications for research and teaching.
2020
Boston, Massachusetts
This theoretical paper explores one way in which student conceptions of substitution equivalence might be classified along a spectrum between operational and structural thinking, which we generalize from Sfard’s theories of the Genesis of Mathematical Objects and research on student conceptions of the equals sign. We provide some sample student work that illustrates different ways in which students may exhibit structural or operational thinking about substitution equivalence. The aim of this paper is to provide an initial testable framework of student thinking around substitution equivalence that could be used as a basis for future studies that confirm, refute, or refine this framework, or use the framework to explore the relationship of different kinds of student thinking around equivalence to other mathematical conceptions and skills.
2020
Boston, Massachusetts
One of the most challenging aspects of doing research in the mathematical modeling genre has been finding an appropriate characterization for the complex interaction of knowledge and cognitive acts that result in coordination of situational referents and mathematical inscriptions. To this end, we introduce the modeling space and illustrate its descriptive and analytic utility.
2020
Boston, Massachusetts
The way instructors respond to student thinking is an important topic of education research, yet a lack of clarity and consistent use of terminology prevents clear communication among researchers on this topic. Researchers use a wide variety of terminology, including teacher follow up, sensitivity to students, uptake, and responsiveness to refer to leveraging student thinking during various instructional practices. In this paper, I use thematic analysis to analyze 34 articles that draw on constructs related to responsiveness to student thinking from within the science and mathematics education literature. Results from this analysis shed light on a distinction between responsiveness as a disposition and as enacted responsiveness. This work has implications for professional development providers interested in supporting instructors in implementing active learning and student-thinking centered practices.
2020
Boston, Massachusetts
Intellectual need is the need that students feel to understand how and why a particular mathematical idea came to be. We are interested in creating tasks that calculus instructors can use to provoke intellectual need. However, the current suggestions for designing such tasks lack detail and don’t account for several issues specific to undergraduate introductory calculus. In this theoretical paper, we discuss the idea of intellectual need, explore three issues related to the teaching of calculus, and present a theoretical model that task-designers can use to frame important factors that affect the development and use of these tasks.
2020
Boston, Massachusetts
Research on Mathematical Knowledge for Teaching has helped the education community understand the complex, knowledge-related factors that shape instructors’ practices and the learning opportunities they create for students. Much of this work has occurred in the context of K-12 teaching. Although expanding, research on knowledge for teaching undergraduate mathematics is not extensive. A similar situation exists in science education. To help support these research efforts and theory development, we analyzed literature on knowledge for teaching STEM content at K-12 and undergraduate levels. Findings take the form of cross-disciplinary themes and descriptions of how components of knowledge for teaching are defined in different disciplines. This cross-disciplinary view into research on knowledge for teaching provides assistance to researchers who wish to leverage work from outside of mathematics and the analysis also reveals some under-examined areas of knowledge for teaching that might be productive foci for future research in RUME.
2020
Boston, Massachusetts
This paper examines the various characterizations of the operational (non-normative) meanings of the equals sign discussed in math education literature. It provides both an exposition and a critique of the various classifications of students’ misunderstandings of the equal sign and of equations. This meta-analysis provides valuable starting points for future research that deepens this active area of exploration.
2020
Boston, Massachusetts
This theoretical report connects an anti-deficit perspective on students’ mathematical sense making with students’ mathematical creativity. The report specifically examines the role of mathematical limitations during the different stages of the creative process. We discuss mathematical creativity in the specific context of constructing everyday examples to explain basis in linear algebra. We consider this construction as a creative activity. We argue that mathematical limitations in students’ initial examples are not only reasonable but can also provide opportunities for more mathematical creativity. While identifying the mathematical limitations led some students to dismiss their examples, thereby ending the creative process, for others it led to further engagement in the creative process through deeper exploration of the everyday example and the mathematics. This report shows that not only does flexibility with full mathematical precision with the examples allow for an anti-deficit interpretation of the students and their sensemaking of the mathematics, such flexibility also sustains the creative process.
2020
Boston, Massachusetts
The purpose of this paper is to expand on Duval's (2017) notion of Cognitive Transformations of Semiotic Representations (CTSRs) and search for CTSRs that go beyond treatments and conversions at a school and entry undergraduate level. Examples of CTSRs that illustrate a progression of mathematical thinking through calculus and real-analysis to topology; and examples from representation theory of finite groups that blends abstract algebra and linear algebra will be explored.
2020
Boston, Massachusetts
This theoretical report discusses metacognition as a tool for connecting teachers’ experiences as learners of undergraduate mathematics to their teaching practice. Academic mathematics has long been considered an essential component of secondary teacher preparation. Current initiatives to make academic mathematics relevant to teacher knowledge and practice focus on course design and the role of instructors. This limits the potential for undergraduate mathematics to impact teacher practice, by restricting attention to select courses. To maximally leverage the impact of academic mathematics on teaching, teachers should be active participants in forming connections. In this report I discuss a metacognitive practice in which teacher are guided to reflect on their experiences as learners of undergraduate mathematics and use these experiences to inform their teaching practice. The report is presented from a theoretical perspective, but it emerged from empirical data. Examples are presented and discussed.
2020
Boston, Massachusetts
Many mathematicians face the unique challenge of acting as teacher educators without necessarily having experience working in secondary education. We propose that examining the learning goals mathematics faculty attend to, as well as how they attend to these goals, provides promising insight into how mathematicians lead content courses for teachers, despite this challenge. In a study which interviewed six mathematics faculty, we were able to categorize the types of learning goals mathematicians seek to address in these content courses. To understand how mathematicians attend to learning goals, we developed an elaboration of the instructional triangle model which reconceptualizes the interactions between mathematicians and students, as well as the environmental influences of these instructional interactions.
2020
Boston, Massachusetts
One expected outcome of physics instruction is that students develop quantitative reasoning skills, including evaluation of problem solutions. To investigate students’ use of evaluation strategies, we developed and administered tasks prompting students to check the validity of a given expression. We collected and analyzed written and interview data at the introductory, sophomore, and junior levels. Tasks were administered in three different physics contexts: the electric field due to three point charges of equal magnitude, the velocity of a block at the bottom of an incline with friction, and the final velocities of two masses in an elastic collision. An unexpected strategy used by some students was to associate physical significance with terms in an expression (e.g., associating a term in an electric field expression with a specific point charge). We explore the significance of these responses and propose explanations for the frequency of this phenomenon in different contexts.
2020
Boston, Massachusetts
As undergraduate mathematics instructors continue to implement active learning strategies and practices, researchers investigate the factors that contribute to classroom environments that are conducive to this approach. In this preliminary report, four “novice” instructors share their reflections, challenges, dilemmas, and personal growth from teaching introductory mathematics courses via an active learning approach. Instructors navigated institutional demands and innovative, “flexible” learning spaces to make reasonable pedagogical decisions. By examining the practicality of their decisions with respect to teaching norms and obligations, this study emphasizes the many resources and supports that instructors utilize to improve their teaching.
2020
Boston, Massachusetts
According to Skemp (1979), a schema is a structure of connected concepts that determines the effectiveness of our director systems. He defines many qualities of a schema, including the strength of connections and the existence of high-order schemas. We use undergraduate-level Topology tasks to investigate two examples of rich schemas to learn more about these two qualities and to see what they may look like practically
2020
Boston, Massachusetts
Mathematical reasoning flexibility across physics contexts is a desirable learning outcome of introductory physics, where the “math world” and “physical world” meet. Physics Quantitative Literacy (PQL) is a set of interconnected skills and habits of mind that support quantitative reasoning about the physical world. The Physics Inventory of Quantitative Literacy (PIQL), which we are currently refining and validating, assesses students’ proportional reasoning, co-variational reasoning, and reasoning with signed quantities in physics contexts. In this paper, we apply a Conceptual Blending Theory analysis of two exemplar PIQL items to demonstrate how we are using this theory to help develop an instrument that represents the kind of blended reasoning that characterizes expertise in physics. A Conceptual Blending Theory analysis allows for assessment of hierarchical partially correct reasoning patterns, and thereby holds potential to map the emergence of mathematical reasoning flexibility throughout the introductory physics sequence.
2020
Boston, Massachusetts
Despite active learning’s promise for student gains across cognitive, affective, and social domains (Laursen et al., 2011), lecture-based teaching dominates instruction in college mathematics courses in the United States (Stains et al., 2018). In this study, college students enrolled in a Calculus I course responded to an inquiry-based learning (IBL) pedagogical approach via a survey that included Likert-scale items and open-ended questions. We sought to determine if there was a relationship among student performance in the class, students’ reported learning gains, and whether or not they would take another IBL mathematics course. Those who responded positively to IBL (49%) performed better on the final exam than those who responded negatively (35%). Additionally, over 55% of those who responded positively reported gains in their ability to work on mathematics collaboratively and communicate mathematical ideas whereas less than 30% of those who responded negatively to IBL reported similar gains.
2020
Boston, Massachusetts
In this report we present the first collection of results of a study of student attitudes towards mathematics during the first year of implementation of a fully active learning in mathematics (ALM) approach in Calculus I at a large, urban, research intensive (R1) institution. Students were randomly assigned to either a treatment section or control section with students in the treatment group participating in the new curriculum. The Attitudes Towards Mathematics Inventory (ATMI) was used to measure student attitudes at the beginning and end of the course and the results compared. The ALM curriculum was found to increase student confidence while the traditional approach decreased it.
2020
Boston, Massachusetts
Previous work by the authors (2019) identified two potential key developmental understandings (KDUs) (Simon, 2006) in the construction of congruence proofs from a transformation perspective for pre-service secondary teachers in an undergraduate geometry course. We hypothesized the independence of the potential KDUs in previous work, meaning that students may have one potential KDU but not the other, and vice versa. We tested this hypothesis with analysis of an expanded data set and found that this hypothesis did not hold in general. We report on the expanded analysis and discuss the implications for the scope and limitation of the potential KDUs. Such work can lead to a more precise understanding of how these potential KDUs may be addressed by instructors of undergraduate geometry courses.
2020
Boston, Massachusetts
In this preliminary report, we present data and initial findings on a preservice teacher’s developing identity of becoming a teacher of mathematics after attending a regional mathematics conference. We are particularly interested in reporting on one case, Krista’s, experiences leading up to the mathematics conference, her experience during the mathematics conference, and how these experiences led to the evolvement of her identity as a preservice teacher of mathematics. Reflections on the conference, focus group data, and autoethnographies written by the students were collected, and we present the preliminary analysis with the overall goals of: discussing the issues/strengths of using autoethnographies as a research methodology in mathematics education, using Krista’s experiences to help recreate similar conference experiences for other preservice teachers, and to discuss how Krista’s identity developed as she became a novice teacher of mathematics.
2020
Boston, Massachusetts
Understanding functions of two variables is difficult even for students who have studied multivariate calculus. The eight participants in this study, prospective middle and high school mathematics teachers, constructed physical models of functions of two variables by creating and assembling sets of transparencies representing cross-sectional planes. The models provided “chunky” representations of surfaces defined by function rules of two variables. This study reports ways in which students with the experience of constructing these models interpreted and operated on a real-world example of multivariate functions (wind chill index) that was reported in a national newspaper. Participants were confronted with potential cognitive conflicts in that the article contained typographical errors that eliminated a variable from the function rule and omitted essential graphical information. First rounds of analysis suggest the importance of focusing on what is being represented rather than solely on the form of the representation.
2020
Boston, Massachusetts
Mathematics education researchers do not yet agree on a shared characterization of proof, or generic proof; neither do mathematicians. Weber's "clustered concept" (2014) isolates six properties that, to personally varying degrees, comprise a valid proof. A generic proof consists of a sufficiently complicated example that illustrates the underlying structures of a generalized argument. Such arguments can be utilized to support student learning of proof. We examine students' and experts' perceptions about proof, explicitly generic proofs. Our findings support Weber's "clustered concept," and extend its utility by showing commensurate properties are also valued by students. In short, we observed that all participants in our study valued an array of psychological and social factors while determining a proof's validity. In addition, our findings suggest that experts do not consider generic arguments to be a proof.
2020
Boston, Massachusetts
This proposal presents data from a session of a design-based research study in which three undergraduate mathematics students compare two attempts to prove that group isomorphism preserves the abelian property. We offer examples of students’ proof construction, comprehension, and validation behaviors in a classroom-like setting. We observe that these three processes alternate and interact in this kind of teaching/learning environment. We argue this space thus provides new opportunities for investigating the relationships among the three kinds of activities to complement previous studies that isolate any one such activity.
2020
Boston, Massachusetts
Student use of mathematics in physics is an area of current interest in RUME and physics education research (PER). In particular, the function concept has been widely studied in RUME but has received less attention in PER. This study probes the ability of introductory physics students to (1) interpret graphical representations of position vs. time functions and their corresponding derivatives, to (2) translate the graphical representation into a meaningful symbolic representation, and to (3) interpret a novel graphical representation. Data were collected through think-aloud interviews and analyzed using a conceptual blending framework (Fauconnier and Turner 1998). The novel representation was challenging for students but in some cases prompted generative reasoning and re-invention of ‘known’ rules and relationships.
2020
Boston, Massachusetts
In recent years, there has a been a push for undergraduate mathematics classrooms to move away from purely lecture to a model where students are more actively engaged in their own learning. Such a transition is hardly a trivial task and requires robust instructional supports. Our recent work endeavors to adapt research-based supports from the K-12 level to the undergraduate abstract algebra classroom. In this report, we share preliminary results from a design-based research project directly aimed at adapting best practices to this new setting. We share several illustrations of how particular teaching routines (Melhuish & Thanheiser, 2017; Teachers Development Group, 2013) can productively unfold in a proof-based setting.
2020
Boston, Massachusetts
Covariational reasoning is a fundamental concept that is necessary to interpret graphical representations of dynamic processes; however, some graphical representations, such as distribution graphs (histograms), are intended to be “read” differently and require an alternative set of strategies for eliciting relevant information and drawing inferences. In this study, semi-structured interviews were conducted with twelve general chemistry students to investigate their reasoning related to the varied population schema, the idea that for a given system, molecules vary with respect to different parameters. Students were prompted to discuss distribution graphs that highlight the variation in a system and their reasoning was analyzed using coordination class theory, a framework that builds on the knowledge-in-pieces perspective to define “concepts” and conceptual change. Preliminary analysis indicates that although students that have productive ideas for sensemaking, they may not necessarily use them appropriately, in some cases unproductively using covariational reasoning to interpret the graphs.
2020
Boston, Massachusetts
Undergraduate learning assistants (ULAs) are becoming more popular in mathematics classrooms. With this growth, there is a need to understand ULAs’ roles in the classroom. Using a four-level distinction from univocal (ULA was the only voice in the conversation) to dialogical (ULA was fostering group discussion), we investigated the discourse between ULAs and students while in an active-learning setting. Our conjecture was that much of the discourse would fall in either univocal or dialogical; however, the ULAs’ discourse was an array of all four ratings. We attribute this to the ULAs recognizing the “intellectual need” of the students in the moment.
2020
Boston, Massachusetts
Linear algebra plays an important role in describing quantum mechanical systems and representing different quantum states. In particular, the ideas of basis and the process of changing basis are fundamental to understanding the nature of quantum states. However, quantum mechanics is abstract and can be very challenging for students. In this paper we describe the development and implementation of an activity that connects a quantum state to a Cartesian coordinate system as an analogy for understanding basis. We further describe changes made from the first implementation as we develop the activity for further use.
2020
Boston, Massachusetts
The derivative is a cornerstone of the first-semester calculus course curriculum. The concepts of function, ratio, slope, variable, covariation, and rate of change are considered to be cognitive roots of the derivative (Larsen, Marrongelle, Bressoud, & Graham, 2017), and this report argues for explicit attention to student understandings of output of a function particularly when considering the derivative in the graphical context. I draw on the location-thinking and value-thinking constructs of David, Roh, and Sellers (2018) for describing students’ thinking about outputs of functions to connect student’s understanding of output to student’s conceptions of differences of outputs. Through a theoretical thematic analysis and a subsequent inductive thematic analysis of clinical interview data with calculus students (Braun & Clarke, 2006), I code the mathematical objects refer to when considering outputs of functions presented in graphical contexts. Codes and potential sources of variation in codes are presented.
2020
Boston, Massachusetts
The Physics Inventory of Quantitative Literacy (PIQL) aims to assess students’ physics quantitative literacy at the introductory level. PIQL’s design presents the challenge of isolating types of mathematical reasoning that are independent of each other in physics questions. In its current form, PIQL spans three principle reasoning subdomains previously identified in the research literature: ratios and proportions, covariation, and signed (negative) quantities. An important psychometric objective is to test the orthogonality of these three reasoning subdomains. We present results that suggest that students’ responses to PIQL questions do not fit this structure. Groupings of correct responses identified in the data provide insight into the ways in which students’ knowledge may be structured. Moreover, questions with multiple correct responses may have different responses in different data-driven groups, suggesting that the both the answer choice and the context of the question may impact how students (implicitly) relate various ideas.
2020
Boston, Massachusetts
Increasing diversity in STEM fields is a moral and economic imperative. However, it is unclear what support systems are most helpful to retaining low-income underrepresented students in undergraduate mathematics programs. This paper and presentation is a preliminary report describing and assessing four support systems developed to increase retention in a four-year mathematics program.
2020
Boston, Massachusetts
There has been a movement away from the pre-requisite model of requiring a year of remedial algebra towards the co-requisite model of providing support in relevant major math courses. The purpose of this project was to examine students’ mathematical beliefs and quantitative reasoning before and after a switch to a co-requisite model.
2020
Boston, Massachusetts
Through participation in a research project on fostering creativity in calculus, two instructors showed shifts in their beliefs on teaching. Participation in the project entailed creating mathematical tasks designed to elicit creative responses from students. Support for task development included participation in weekly online professional development sessions. In this paper, we share one instructor’s shifts in beliefs as well as alignment of her pre-existing beliefs with pedagogical actions. Preliminary analysis of her entrance tickets to the professional development sessions and her exit interview indicates that this instructor a) shifted her previous beliefs about a perceived time pressure and b) manifested her existing beliefs into actions regarding multiple-approach tasks.
2020
Boston, Massachusetts
Philosophers, mathematicians, and students struggle with the notion of infinity. This study investigates one undergraduate student’s response to tasks involving infinite iterative processes in different contexts. A semi-structured clinical interview was conducted to describe the types of mental structures the participant had constructed to solve the proposed tasks. APOS theory was then successfully able to predict later student’s responses. This study also suggests that how students depend on time to understand infinity and infinite processes may be an area for further investigation.
2020
Boston, Massachusetts
When using function notation, some students describe f(x) as meaning the same thing as the variable y to represent the output value of the function. This paper describes interviews with two students with vastly different interpretations of function notation. The interviews were designed to explore how students interpret function notation when it is used to represent the output for a given input.
2020
Boston, Massachusetts
Covariational reasoning — how one thinks about the way changes in one quantity affect another quantity — is essential to calculus and physics instruction alike. As physics is often centered on understanding and predicting changes in quantities, it is an excellent discipline to develop covariational reasoning. However, while significant work has been done on covariational reasoning in mathematics education research, it is only beginning to be studied in physics contexts. This work presents preliminary results from an investigation into expert physicists’ covariational reasoning in a replication study of Hobson and Moore’s 2017 investigation of covariational reasoning modes in mathematics graduate students. Additionally, we expand on this work to include results from a study that uses slightly more complex physics-context questions. Two behavioral modes were identified across contexts that appear distinct from those articulated in the Hobson and Moore study: the use of compiled relationships and neighborhood analysis.
2020
Boston, Massachusetts
Self-efficacy (Bandura, 1977), an individual’s belief in his or her ability to succeed at a specific task, is a predictor of student performance and persistence in mathematics (Pajares & Miller, 1994; Zeldin & Pajares, 2000). Thus, it is important to understand how students’ self-efficacy changes in different settings. When designed carefully, certain mathematics learning environments are more conducive to students’ development of self-efficacy as they allow for multiple self-efficacy opportunities (Sawtelle, Brewe, & Kramer, 2012). Flipped classrooms (Lage, Platt, & Treglia, 2000) reverse classroom lecture and out-of-class assignments and may increase self-efficacy, as students have opportunities for collaborative work and instructor feedback during class. The purpose of our study was to investigate changes in students’ self-efficacy in a flipped Calculus II course. Quantitative findings included significant increases in students’ self-efficacy in Calculus. Qualitative findings revealed that students believe their experiences in a flipped classroom setting increase their mathematics self-efficacy.
2020
Boston, Massachusetts
Adopting open educational resources (OER) with an organized educational frame has the potential to not only reduce student cost for learning materials but also improve student learning, instructional methods, and educational environments. The aims of this study are to explore student perspectives on the use of an OER platform in blended learning, and to examine student achievement, engagement, and opportunity for learning mathematics. As a mixed project, the data was collected from 423 students in 15 sections of Elementary Statistics during four semesters. The results of this study showed that the use of an OER learning platform in blended learning promoted student engagement and significantly increased student opportunity for learning. There were not significant differences in student achievement between adopting an OER learning platform in blended learning and adopting commercial resources in regular classes. This study will contribute to the knowledge of the open educational use of technology.
2020
Boston, Massachusetts
This article describes a quantitative research project with the sole aim of exploring the factors affecting learning outcomes of students in higher education mathematics. The chosen factors are mainly from students’ personal constructs which are approaches to learning, self-efficacy and prior knowledge. Results of analysed two sets of data consisting of 234 engineering and economics students as well as 253 engineering students that offered a first-year mathematics course are reported. The preliminary findings established a differential classification of the students’ approaches to learning into deep and surface. It has also exposed the relationship between calculus self-efficacy and approaches to learning with higher self-efficacy students identified with deep approaches and lower self-efficacy students identified with surface approaches.
2020
Boston, Massachusetts
This paper examines the mathematical activity of two students as they engage in reasoning about topics in ring theory by analogy with topics in group theory. Analogies are represented by mappings between a source and target domain (Gentner, 1983). Participants were given task-based interviews, each focused upon a particular structure from ring theory: rings, subrings, ring homomorphisms, and quotient rings. Techniques from grounded theory were utilized to analyze the interview transcripts with the goal of interpreting and describing the mathematical activity of the students. Preliminary results indicate that there are three main categories of activity: foregrounding the source or target domain, focusing on similarities or differences, and which mapped aspects of the domains are the focus of the reasoning. “Pathways” are identified as a way to capture the dynamic nature of students’ reasoning by analogy in abstract algebra.
2020
Boston, Massachusetts
A comprehensive graduate teaching assistant (GTA) training program in mathematical sciences designed at one institution is being replicated at two peer institutions. This paper presents the findings of a baseline comparison of the three universities undertaken at the start of the project to inform its adaptation and implementation at each institution and the evaluation of its impact. Program components include a first-year teaching seminar, peer mentoring and support from a peer TA coach, a Critical Issues in STEM Education seminar, and K-12 outreach to inform understanding of the pipeline. Differences in undergraduate demographics and performance in introductory mathematics courses, GTA responsibilities, prior departmental GTA training elements, and GTAs attitudes towards teaching mathematics/statistics are presented. Implications for program implementation and assessment of study goals related to institution differences are presented.
2020
Boston, Massachusetts
The term computational thinking has engaged multiple disciplines in discussions about how to prepare students for careers in a technological society. However, a wide variety of definitions and settings has made it difficult for researchers to clarify what is meant by computational thinking. In this paper, we present a case study that explores one setting in which computational thinking and mathematics interact, through an interview with a computational mathematics PhD student. We consider the influence that computational thinking and mathematical knowledge have on each other through three key moments in the student’s dissertation process. Additionally, we consider the role of affect in computational thinking and the potential benefits that computational thinking carries for the future of mathematicians.
2020
Boston, Massachusetts
Precalculus to single-variable calculus (P2C2) courses are a critical first educational step for students pursuing any science, technology, engineering, and math (STEM) degree. As students come into university courses with a wide range of preparation and skill, institutions need to be ready to meet their needs. One way this issue is being addressed is by the development of different course variations in the P2C2 courses. In this study we focus specifically on what motivates the creation of such course variations and how their success is perceived. Using open-ended survey responses from key faculty and instructors at ten institutions we identify themes relating to both the motivation and perceived success of course variations.
2020
Boston, Massachusetts
This article is a preliminary report describing a mathematics outreach program, which is a partnership among university faculty, underrepresented university students, and elementary school students in the west coast. This program aimed to develop math literacy through after-school engagement by providing opportunities to university students to apply university-learning experience in an out-of-class setting. The study participants were four university students who taught various mathematical topics to about twenty-five 3rd to 6th graders weekly for five weeks at a local public elementary school. The data consisted of university students’ self-reflections, which revealed commonalities between their perceptions of the learning objectives for elementary students and for themselves that they thought were met. They also developed an awareness and understanding of various issues that exist within education; they learned new mathematical vocabulary, and made explicit connections between their teaching experience and university-learning experience.
2020
Boston, Massachusetts
The purpose of this qualitative study is to examine the beliefs and practices of undergraduate tutors and graduate teaching assistants who work in mathematics support centers. This report is situated in a larger study that is being conducted at two four-year institutions in the U.S. and focuses on our preliminary analysis of six graduate teaching participant responses from one institution. For the first phase of the study, we asked participants to complete a survey that was based off of the Teacher Beliefs Interview protocol (Luft & Roehrig, 2007) in order to examine their beliefs and practices before they participate in mathematics tutor-specific professional development activities. Using a modification of Luft and Roehrig’s coding scheme, we coded survey item responses as Instructive, Transitional, or Adaptive. These codes gave us a snapshot of their beliefs and practices around tutoring in mathematics
2020
Boston, Massachusetts
Linearity plays an extensive role in both elementary and more advanced mathematics. Unlike other such topics, investigation into student understanding has been limited. Nineteen students enrolled in multivariable calculus were asked about their conceptions of linearity. Most provided only elementary notions. Implications and future research are considered.
2020
Boston, Massachusetts
While many aspects of the teaching and learning of advanced mathematics have been explored, the role, construction, and values of homework has been virtually ignored. This report draws on task-based interviews with six mathematicians to explore the relationship between an instructor’s learning goals and the factors they consider in selecting homework problems as well as general homework construction heuristics. Our initial findings are that 5 of the 6 participants viewed homework as a critical part of student learning, that the majority of the participants’ claims focused on either the mathematics or how the problem would help students learn, and that there was variance among the mathematicians in terms of selection of problems and rationales.
2020
Boston, Massachusetts
In this study, we examine two inquiry-oriented classes using Battey and Leyva’s (2013) relational interactions framework. We examine the relational interactions that occurred in a class with gender-equitable student learning outcomes and a class with gender-inequitable student learning outcomes. Although we expected to see differences in these two classes, results show that both instructors had similar relational interactions, focusing mostly on acknowledging student contributions. However, in one class there were noticeable negative relational interactions with women students that might help us better understand the gender differences in the classes.
2020
Boston, Massachusetts
Prospective mathematics teachers are often required to take a course in abstract algebra. However, there is still some question as to how prospective teachers connect their knowledge of abstract algebra to secondary algebra and how they might draw upon this knowledge as they teach. This study explores a prospective teacher’s conceptions of inverse functions and their relation to the algebraic group of invertible functions under composition. It explores how this prospective teacher draws on her mathematical knowledge as she responds to student thinking.
2020
Boston, Massachusetts
Calculus is about change and thus dictates a need for instruction involving dynamic imagery. DIRACC (Developing and Investigating a Rigorous Approach to Conceptual Calculus) utilizes animations to support students’ dynamic imagery. This paper investigates how students use and understand animations in the DIRACC textbook in connection with associated calculus topics.
2020
Boston, Massachusetts
This study investigates how student engagement with a self-regulatory activity is linked to subsequent student performance. College Algebra students were given an activity that was focused on the planning, monitoring, and evaluating aspects of self-regulation within the context of completing the square. Responses on this self-regulatory activity were compared to performance on completing the square problems on exams. We discuss cases of an engaged, a moderately engaged, and a disengaged student and link to exam performance. Further, we share how findings are impacting future revisions of this self-regulation activity.
2020
Boston, Massachusetts
In this preliminary report we discuss the development of a framework regarding the facilitation of online working groups geared at supporting instructional change at the undergraduate level. The research in undergraduate mathematics education includes large-scale projects aimed to support individuals, departments, and the mathematics community in reforming their instruction to align with recommendations from professional organizations and align with existing mathematics education research standards. One avenue that needs attention is the use of online synchronous environments to match faculty across the world and form collaborations to support the inclusion of student-centered activities in our mathematics classrooms. An important part of that work is to understand how to facilitate those online synchronous environments. This preliminary report discusses the actions that facilitators take in these environments and lays the groundwork for the use of this framework in our and other contexts going forward.
2020
Boston, Massachusetts
As support for non-lecture pedagogy grows, the number of instructors implementing active-learning approaches in their undergraduate mathematics classrooms is steadily increasing. The proliferation of such techniques has led to the inevitable investigation into the potentially differential effects for various sub-populations and equity therein. In this study, the primary objective was to link specific affective factors to student performance with an eye to variations by gender and/or pedagogical approach. Results suggest that different forms of instruction impact men and women in different ways; specifically, affective reports are better predictors for women’s scores on a content assessment than for men and this effect is more salient in lecture classes as compared with active-learning environments. Furthermore, there is evidence that which specific affective traits influence performance also varies by subgroup.
2020
Boston, Massachusetts
One purpose of introductory statistics courses is to acquaint students with important disciplinary themes, including the nature of a statistical investigation and the types of activities that data analysts engage in. In this paper, we discuss the development of a survey instrument intended to assess the disciplinary perspectives of introductory students. Building from disciplinary profiles proposed in our previous work with social science students taking statistics, combined with current discussions in the literature, we have constructed a four-dimension framework to represent various perspectives we believe students may hold. Through a series of think-aloud interviews with eight, first-semester college freshmen, we focused on establishing the construct validity for items designed to assess students’ views about statistics. This paper documents our methods, highlights our interview findings, and outlines our next steps for validation.
2020
Boston, Massachusetts
The first law of thermodynamics is an essential scientific principle with broad relevance across the fields of STEM. As a foundational principle, the first law provides students with conceptual and mathematical tools to solve cross-disciplinary problems. The pilot study summarized in this proposal investigates the ability of engineering students to address first law problems across the field-specific contexts of chemistry, engineering, and physics. Preliminary results suggest that the students’ prior experiences in a calculation-intensive thermodynamics course biased student reasoning in favor of solving problems numerically. The application of transfer of learning frameworks to this data set reveal potential conceptual and epistemological resources activated by students that may explain this observed tendency. These results have implications on the application of calculation-intensive problems in science and engineering coursework.
2020
Boston, Massachusetts
This study is a post-hoc analysis of pre and post survey data from two rounds of a mathematical modeling competition comprised of high school and undergraduate participants (n=107). The purpose of this study is to describe the expectations participants held going into the competition and compare them to those held by researchers and designers of modeling competitions. Additionally, this study examines participants’ satisfaction with the competition. Results showed that participants, researchers, and designers held differing expectations. Participants expected to gain more practical experience but afterward reported gaining practical experience and soft skills. Lastly, participants tended to gain what they expected from the competition; those who felt their expectations were not met would not recommend the competition to others (regardless of what their initial expectations were)
2020
Boston, Massachusetts
Graduate teaching assistants (GTAs) play an important role as instructors in math departments. As a result, professional development (PD) opportunities have been developed to support these primarily novice instructors, but these programs can vary widely. This study assesses the state of a PD program for GTAs at a large, public, doctoral-granting institution. Surveys and interviews were analyzed with using Social Cognitive Theory (SCT) as a framework.
2020
Boston, Massachusetts
Although mathematical confidence is known to relate to students’ level of mathematics, the relationship between a student’s content-specific confidence and the student’s familiarity with the content remains unexplored. Using a regression analysis of survey data from a single institution, this study examines this relationship for three common, lower-division, undergraduate mathematics courses. Students in the targeted courses reported high levels of content familiarity and confidence they could do the work for the class without instructor intervention. Familiarity was the biggest predictor of students’ content-specific confidence, but some orientations towards mathematics also mattered. When tested, students who reported they could do problems were not highly accurate in their assessment. Taken together, these findings suggest that there might be a benefit in helping students better assess their content knowledge early in their courses.
2020
Boston, Massachusetts
My work, an ongoing research, focuses on the identification of features of discourses in describing linear independence, borrowing ideas from Sfard (2000; 2001) and Presmeg (1997;1998). This paper reports the analysis of two participants’ discourses revealed in their responses to a single question. The findings showed that previously formed templates and signified-signifier pairs facilitated new semiotic spaces from which new signifiers (with new meanings) were adopted. Furthermore, participants’ descriptions of linear independence closely resembled features of their templates
2020
Boston, Massachusetts
As part of a large grant that investigates how students and instructors use university open source, and open access textbooks in teaching, we collected learning and performance data from students. In this paper we explore the relationship between textbook format (PDF versus HTML) and student outcomes in three undergraduate mathematics courses—linear algebra, abstract algebra, and calculus. We did not find differences in student outcomes by textbook format or course. We propose some reasons for these findings including little differences in instructor use of the textbook formats. We pose some questions for the audience.
2020
Boston, Massachusetts
We report on a teaching intervention in an ordinary differential equations (ODEs) course for engineering students focusing our attention on the role of nonstandard problems in the development of students’ conceptual understanding. The lecturer designed a set of problems challenging traditional approaches to Existence and Uniqueness Theorems (EUTs). Our analysis of the students’ mathematical discourse developed in the process of group discussions within Commognition Theoretical Framework reveals unresolved commognitive conflict. We suggest how modifications in the problem design prompted by the students’ mathematical discourse can be used to improve students’ conceptual understanding of the material.
2020
Boston, Massachusetts
Teachers’ mathematical meanings impact their instructional practices and constitute their images of the mathematics they teach and intend students to learn (Thompson, 2013). While numerous studies have focused on K-12 teachers’ mathematical knowledge for teaching (MKT) and mathematical meanings for teaching (MMT) (Thompson, 2013), few studies have examined university level instructors’ mathematical meanings (e.g., Musgrave & Carlson, 2016). In this report, we explain what we mean by mathematical meanings for teaching and productive student-teacher interactions and use video data to characterize the relationship between teachers’ MMT and decentering actions when teaching. Our results illustrate how a teacher’s MMT can influence the teacher’s ability to make sense of and use student thinking during an interaction in which the teacher is attempting to decenter. Conversely, we illustrate how decentering actions can lead to advances in a teacher’s MMT.
2020
Boston, Massachusetts
While videos are becoming more pervasive in math instruction and remediation, little is known about what students are doing to make sense of the math while watching videos. This study presents a case of one student who is using videos to learn college algebra. Using the didactic contract, we suggest that the student’s inability to live up to the rules which she has for herself and the video may impact her mathematics learning while video watching.
2020
Boston, Massachusetts
The goal of this study was to determine the ways in which comics affect student learning in undergraduate math courses, specifically in first-year linear algebra. Students had access to eight comics, posted roughly each week during the term. The survey data collected shows that the majority of respondents reported a better understanding of the material after reading the comics and that the comics had a positive impact on their engagement, attitude, motivation, and overall understanding.
2020
Boston, Massachusetts
The presented project combines two core concerns of the authors: University teaching is repeatedly confronted with the problem of very large learning groups and therefore poor individual support. To address this problem, the authors have developed a technical system that automatically answers questions from learners in a statistical introductory lecture. The usage data of this system can be used for a second purpose. Combined with further usage data from other systems and quantitative surveys, it can be determined what influence subject-specific attitudes and beliefs have on learning behavior. The poster presents the technical solution, its educational embedding, results of the evaluation and first insights of the learning analytics
2020
Boston, Massachusetts
This poster focuses on the first stage of a study designed to explore how preservice teachers establish connections between abstract algebra and secondary school mathematics. Additionally, I will describe the motivation for collecting the data in the context of the larger research study. The purpose of this first stage was to gather information regarding mathematical connections between abstract algebra and secondary school mathematics, through the use of online surveys, from the perspectives of mathematics faculty as well as practicing secondary mathematics teachers. The survey data was analyzed using qualitative methods. Many mathematics faculty reported that they incorporate mathematical connections to secondary mathematics into their abstract algebra instruction.
2020
Boston, Massachusetts
In this poster we present a newly implemented program of introductory mathematics classes at Sonoma State University. We illustrate the new model itself and the methods we use to assess student learning outcomes. We summarize some initial findings from the assessment and welcome ways to learn and adapt our methods for more informative assessment.
2020
Boston, Massachusetts
Researchers are not paying attention to first, students’ engagement and secondly how gender identity shapes engagement experiences in undergraduate mathematics classrooms. This study investigates student engagement and gender identity while learning mathematics using a mobile app that collected student engagement reported by students. This paper discusses preliminary results on students’ engagement and gender identity in undergraduate mathematics classroom. Those identifying as women reported higher engagement than those identifying as men.
2020
Boston, Massachusetts
A post-secondary credential continues to be a goal of many people in the United States, not only for self-esteem but for career purposes. Students are often required to take remedial mathematics as they are unprepared for college-level coursework. Results of a regression analysis are presented in this poster comparing factors that might be influential in the need for mathematics remediation for three racial groups: White, African American and Hispanic students.
2020
Boston, Massachusetts
Undergraduate students pursuing STEM careers have varied perceptions of how mathematics is used in their careers. A survey of community college students enrolled in College Algebra measured interest in mathematics and interest in STEM-related careers, as well as a student’s perception of how mathematics was used (or not) in their chosen career. Analysis revealed that students’ beliefs related to the usefulness of mathematics in their chosen career predicted their interest in mathematics in general, and their interest in STEM careers in some cases.
2020
Boston, Massachusetts
In this research we explored undergraduate students’ geometric reasoning of complex integration before the instructor taught it in the course. We found that the participants leveraged their knowledge about complex multiplication to construct a local model of complex integration. In an effort to develop a global model, which attends to the accumulation aspect of integration, we found that our participants struggled due to the thinking real, doing complex phenomenon.
2020
Boston, Massachusetts
With recent recommendations of the MAA, instructors of undergraduate math courses are encouraged to incorporate more active learning strategies into their teaching. While achievement-focused studies consistently show that the use of active learning strategies in undergraduate STEM instruction is better than lecture alone, little research has been done on the student experience with these active learning strategies. In addition, while improving diversity in STEM fields is an oft-stated goal of reform in undergraduate math instruction, few studies have looked at how different populations might experience the same active learning classroom differently. This poster presents the methodology and early results from a study that in is an effort to address both of these gaps. The study uses in class group observations and follow-up interviews to develop an understanding of how gender identity and/or sexual orientation might influence student experiences in small groups in undergraduate math classrooms.
2020
Boston, Massachusetts
It is well known that undergraduate calculus students struggle with the concept of derivative and ideas related to it. Yet, there is less research available on more specific techniques and applications of the derivative. The goal of this study was to examine student understanding of, and ability to carry out one such technique, implicit differentiation. Data was collected through both written surveys and clinical interviews. Findings suggest that students do not have a strong understanding of implicit differentiation and that their difficulties with computational problems stem from a variety of issues they have related to functions.
2020
Boston, Massachusetts
Reaction Coordinate Diagrams (RCDs) aid in the visualization of the thermodynamic and kinetic factors which influence chemical reactions. These graphical representations pose unique challenges for chemistry students given the abstracted physical dimensions that define the Cartesian coordinate system. General chemistry students’ interpretations of RCDs were investigated by applying mathematics and chemistry education research frameworks in a semi-structured interview setting. Findings suggest that students’ interpretations of the points and trends along these diagrams strongly influence the physical meaning they attribute to RCDs.
2020
Boston, Massachusetts
Students appear to experience difficulty in coordinating different types of reasoning and different types of representations. Furthermore, they appear to start integrating these different modes of reasoning or representations when prompted to embody a given abstract mathematical concept. This research is part of an ongoing investigation into how students reason geometrically about various aspects of complex numbers, such as derivatives of complex-valued functions or the Cauchy-Riemann equations (hereafter the CR equations). Collection and analysis of the data is ongoing but is hoped to produce potential learning trajectories or natural ways in which students reason about the CR equations and their connections to the derivative.
2020
Boston, Massachusetts
Mathematical models are ubiquitous in science and science education and serve as tools for constructing explanations and making predictions about real world phenomena. However, undergraduate science students often use mathematical models algorithmically, without consideration of the physical meaning of mathematical variables. To support chemistry students’ ability to engage in “meaning making with math,” we developed a series of instructional activities which lead students through using, evaluating, and revising mathematical models of chemical phenomena. Here, we discuss development and refinement of an activity focused on gas behavior and discuss theoretical frameworks that guide our activity design.
2020
Boston, Massachusetts
The Mathematical Inquiry Project (MIP) is an NSF-funded collaboration of mathematics faculty from all 27 public institutions of higher education in Oklahoma to support inquiry-based learning in entry-level mathematics courses. This poster reports on the initial development of a Community of Practice, including shifts in the participants’ conceptions of the joint enterprise and their identities in relation to it.
2020
Boston, Massachusetts
The poster presentation describes the work undertaken in a mathematics methods course. 7 pre-service teachers (PSTs) critiqued mathematics tasks for their cognitive demand. Following critique of the tasks they planned for implementing and then implemented the tasks. Their critique of the tasks allowed them to learn about the properties of effective mathematics tasks. Further, they began to see the role teachers play in maintaining or decreasing cognitive demand.
2020
Boston, Massachusetts
In this poster, we provide an overview of a design-based research project investigating student discussions surrounding proof in abstract algebra. This project aims to incorporate best practices for orchestrating discussions developed within the K-12 spectrum into the context of advanced mathematical proving activity. We describe three aspects of the project: general task design, task implementation, and future work to be considered based off the implementation.
2020
Boston, Massachusetts
The difficult transition from computational mathematics to advanced mathematics is well-documented in the research literature. The ASPPMIRE project has two main goals. The first is designing curriculum materials that support students in reinventing fundamental concepts in abstract algebra and real analysis, along with fundamental proof skills. The second supports instructors in implementing inquiry-oriented instruction in their own classrooms. The purpose of our poster presentation is to invite the research community to discuss our project’s motivation, goals, and study design.
2020
Boston, Massachusetts
Proof by mathematical induction is known to be conceptually difficult for undergraduate students. We present a model that may simulate the impact of logical implication on students mastering proof by induction. We combine Piaget’s action-object theory of mathematical development with a psychological model of working memory and Harel and Sowder’s proof schemes. We analyzed three sets of written assessments from two Introduction to Proofs classes: after students learned about logical implication; before and after instruction on proof by induction. We examine the relationship between proficiency with mathematical induction and treating logical implication as an object within these two classes.
2020
Boston, Massachusetts
McLoughlin and Droujkova (2013) developed a diagrammatic definition of multiplication that uses parallel lines to continuously scale the length of one segment by the length of a different segment. This contemporary treatment of the constructability of products (and quotients) is potentially significant for the undergraduate mathematical preparation of pre-service elementary teachers, who tend to conceptualize multiplication in terms of repeated addition. We take up here the design challenge of constructing a physical tool that models multiplication as a continuous scaling operation as opposed to a repeated grouping operation. We ask: How can the parallel shadows interpretation of real-number multiplication be used to design a physical tool?
2020
Boston, Massachusetts
We present the methodology and preliminary findings from a study whose purpose is to uncover student thinking around volumes of solids of revolution, with a specific focus on how students create and use visual images. We use Gutierrez’ visualization framework (1996) to analyze how students use representations of their mental images to solve tasks. Initial findings suggest that students struggle representing slices that can be used to measure the volume of a sphere; first attempts instead make use of two-dimensional figures.
2020
Boston, Massachusetts
Identifying formative experiences that affect mathematical problem-solving (MPS) development and trajectories for rising mathematicians presents a complex challenge. Despite the existence of various descriptions of isolated MPS behaviors for populations of mathematicians of different levels of expertise, there is still insufficient understanding of how MPS develops as a result of extended mathematics education. This poster presents preliminary findings of a study which aims to characterize formative experiences of advanced undergraduate mathematics majors and first-semester mathematics graduate students which, in interviews, they identified as affecting their MPS strategies or disposition towards solving difficult mathematics problems. We also aim to gather feedback on study design and potential directions for analysis of the interviews.
2020
Boston, Massachusetts
Students ask the question “when is this ever going to be useful?” when speaking about mathematics. If we take this as a question about meaningfulness, how can teachers respond through their instruction (if they choose to do so) and how do they even understand the terms ‘meaningful’ and ‘meaning’? Prior research suggests beliefs influence instruction. I am interested in understanding how beliefs then influence what meanings instructors’ goals and instruction focus on. This theoretical paper aims to synthesize prior research to elaborate a framework of meaning orientations for instructors’ goals. Future research can then look at how and why instructors focus on certain orientations of meaning over others.
2020
Boston, Massachusetts
This exploratory study investigates how students attend to conflicts that arise during the validation of a model while engaging with a mathematical modeling task. Analysis of a one-on-one task-based interview revealed that the student attended to conflict in the following ways: (1) modifying the object of validation (2) modifying the standard of validation, and (3) leaving the conflict unresolved.
2020
Boston, Massachusetts
Assessment is a central issue to the lives of educators and students. The purpose of assessment is to certify achievement or to facilitate learning (Boud, 2000). At all levels, from programmatic to course-specific, we strive to assess in a way that accurately measures achievement or progress relative to one’s goals. In this poster, we describe our experiences assessing students in an undergraduate mathematics course focused on creativity in mathematics. While traditional forms of assessment often focus on repeating mathematical procedures and demonstrating abilities of routine skills (Firestone, Winter, & Fitz, 2000; Lesh & Clarke, 2000), we attempted to align assessments with the alternative instruction we employed. Here we describe our assessments and report on preliminary findings of students’ reactions to the assessments through reflective journal entries, focus group interviews, and pre- and post-course surveys, outlining a need for revisions to traditional assessments.
2020
Boston, Massachusetts
The purpose of this study is to investigate student thinking in the context of graphs of functions of two variables. The data presented here is an example of the ways of thinking students exhibit when working with graphs of multivariable functions.
2020
Boston, Massachusetts
This poster discusses geometric definitions that impact the teaching of college geometry.
2020
Boston, Massachusetts
Promoting equity in undergraduate mathematics education is of vital importance, yet has received considerably less attention than equity in K-12 mathematics. The current study focuses on a pedagogical training program for graduate teaching assistants’ (GTAs), which emphasizes equity in their teaching of undergraduates. The study examines GTAs’ journals and open-ended survey responses, including their definitions of equity and the ways they promote equity in their classrooms. The research will foster discourse about ways of promoting equity in undergraduate mathematics and about professional development for undergraduate mathematics instructors.
2020
Boston, Massachusetts
As our RUME community and attendance at our national conference grows, it is vitally important that we continue to attend to issues of equity and inclusion. In this poster presentation, we discuss how we can use regional RUME conferences to broaden participation and support inclusion in the larger RUME community. Using the lens of Lave and Wenger’s (1991) communities of practice and legitimate peripheral participation, we analyzed survey responses from individuals who either attended or expressed interest in attending a regional RUME conference. We found that local RUME conferences provide newcomers and novices with the opportunity to (1) learn more about RUME, (2) network with RUME participants, including experts/old timers, and (3) develop professionally as both teachers and researchers.
2020
Boston, Massachusetts
Throughout the past few decades, the term active learning has been used to describe a variety of classroom instructional techniques and pedagogy. In this poster, we explore the graduate teaching assistants’ conceptualization and implementation of active learning strategies at the start of a funded project evaluating a multifaceted GTA training model.
2020
Boston, Massachusetts
In this work we describe a study that analyzes student engagement in the process of sensemaking, which involved students developing metamodeling ideas related to mathematical models in chemistry, that is, ideas regarding the nature and purpose of models. This project defines sensemaking as the process of constructing and evaluating explanations to address an apparent inconsistency (i.e., “figure out” a problem). This process involves students using prior knowledge in combination with provided information to resolve a gap in understanding. For our dataset, students collaboratively worked through activities designed using the learning cycle (Process Oriented Guided Inquiry Learning, POGIL), which involves student-led movement through stages of exploration (direct questioning), concept invention (developing a formal definition for an idea), and application (using students’ constructed concept in a new context). Preliminary results from this qualitative study emphasizes how students construct explanations and engage in metamodeling ideas as part of the sensemaking process.
2020
Boston, Massachusetts
This poster discusses a sequential, mixed-methods study of an established program that incorporates active learning within recitation sections to support the academic performance and retention of students from underrepresented groups at a large research university. The study examines academic performance in math courses, degree attainment, and overall experience of students who participated in the program.
2020
Boston, Massachusetts
We present preservice teacher’s (PST’s) progression of thinking during the Ant Farm Task (AFT). We describe the PST’s construction of the Cartesian plane in four critical phases. The AFT presented us an opportunity to identify important cognitive activities that supported the PST’s construction of the Cartesian plane.
2020
Boston, Massachusetts
Recently group testing in mathematics classes have been explored for their equitable use as a means to re-humanize mathematics. This study extends this discussion by examining interactions within differently composed groups to identify group structure for equitable assessments.
2020
Boston, Massachusetts
The purpose of this pilot study was to identify quantitative reasoning (QR) skills 9th grade high school students need to successfully construct evidence-based scientific explanations. Knowing what kinds of QR skills 9th grade students need in science classrooms can help inform pre-service teacher programs, since many of these skills are taught in mathematics classrooms. Preliminary results indicate that in order to create a claim, construct evidence about the claim, and successfully reason about the claim and evidence, students need to (a) contextualize the variables, (b) be open-minded for any relationship in the data, and (c) use quantitative language.
2020
Boston, Massachusetts
The question of what teachers need to know about their subjects to be able to promote powerful and flexible knowledge and understanding among their students has been difficult to answer. This difficulty has been a result of the many different conceptualizations of teacher knowledge, which until now have mostly been general and not domain specific enough. In an effort to promote a shift from these, this study used the Knowledge of Algebra for Teaching (KAT) project’s conceptualization of knowledge for teaching algebra and the instruments developed from that project. Factor analysis of data collected has not only corroborated the KAT project’s hypothesized knowledge but also pointed to a need to reconsider some of the assumptions made in the KAT framework. As a result, a new conceptualization of knowledge for teaching algebra that allows teacher knowledge to be assessed in measurable terms will be discussed.
2020
Boston, Massachusetts
This study investigates how spatial diagrams are used by pre-service elementary teachers to construct arguments about measures of solid figures. Dynamic spatial diagrams offer immersive three-dimensional representations of three-dimensional geometric figures, where learners can take perspectives are not accessible in two-dimensional representations. The results describe how PSETs used perspectives outside and within a dynamic spatial diagram to make arguments.
2020
Boston, Massachusetts
Many undergraduate students experience significant difficulty in learning to prove mathematical propositions. In contrast to previous work focusing on the final products of proof, this work aims to characterize proving processes of undergraduate STEM majors who have recently completed an Introduction to Proof (ITP) course. This study extends Carlson and Bloom’s framework in order to better account for how undergraduate students navigate the potentially multiple “stuck points” that students encounter during their proving processes.
2020
Boston, Massachusetts
We conducted a constructivist teaching experiment to better understand how introductory calculus students’ units coordinating activity supports their learning of productive conceptions of rate of change. This poster serves to illuminate what we learned while supporting Rick, a student assessed as assimilating with two levels of units.
2020
Boston, Massachusetts
We use activity theory to analyze tensions manifested in an activity system of an extra-curricular mathematical modeling (MM) project with biology undergraduates at a research-intense Scandinavian university. We present the evidence that the use of MM in education of biology students accentuates conceptual understanding of mathematics and may lead to beneficial shifts in pedagogical practice.
2020
Boston, Massachusetts
This paper presents a pilot investigation into the effects of stereotype threat on women’s performance in upper-level undergraduate mathematics and potential influences of the affective factors, mindset and sense of belonging. In contrast to Good, Aronson, & Harder’s (2008) findings, results indicate that the women in this study did not seem to be adversely affected by stereotype threat. Furthermore, sense of belonging to the domain of mathematics was found to be a statistically significant predictor of performance on a mathematics assessment.
2020
Boston, Massachusetts
This research describes results from both curricular and structural reforms in introductory math courses at Michigan State University (MSU), a large four-year public university. These reforms have focused on providing students with new pathways to success by eliminating an intermediate algebra course that had a low pass-through rate, did not count toward graduation requirements, and enrolled disproportionately high numbers of racial and ethnic minority students. The reformed curricula now place students directly into either quantitative literacy-focused courses for students on a degree pathway not requiring calculus or a reformed college algebra course sequence for students on a degree pathway that does require calculus.
2020
Boston, Massachusetts
To support mathematics graduate student instructors (GSIs) as teachers, a collaboratively generated curriculum has been developed for all precalculus courses at the University of Nebraska—Lincoln (UNL), focusing on problem-based, student-centered instruction including lesson plans to actively engage students. Over the past few semesters, GSIs and a faculty member at the University of South Carolina (USC) have been working to adopt these materials at their institution for transformational change in GSI teaching. We analyze the allocation of time and teacher design practices for various tasks associated with the adaptation of problem-based handouts and lesson plans. We also consider the implications for personnel with a vested interest in implementing similar changes.
2020
Boston, Massachusetts
On a metacognitive level, reflection is known to be an essential skill for improving learning. In practice, students may not always use it of their own accord to improve this kind of learning because it can be mentally demanding, and they often believe that learning mathematics means getting the right answer without reflecting too much on their learning. To encourage reflection on understanding of concepts in my Precalculus course, I introduced assignments that required students to reflect and self-assess their learning of concepts. This study explores this type of pedagogical approach.
2020
Boston, Massachusetts
This poster presents evidence and results from two of the codes, Instructor-Student Continuum of Instruction and Classroom Environment, from the current version of our video coding protocol for community college algebra instruction. We highlight the findings, teaching examples, challenges, and implications from analyzing 135 hours of video data from 44 community college algebra classrooms, entailing coding of 1110 segments for teaching algebra at the community college level.
2020
Boston, Massachusetts
This poster shares a case of reconstructive generalizing. Specifically, Jolene’s response to a reconstruction task is presented as an illustration of an approach to reconstructive generalizing. Given a generalization and a request to develop a new generalization about an expanded domain, Jolene developed a justification for the given generalization and used that justification to scaffold her reasoning toward a reconstructed claim.
2020
Boston, Massachusetts
This interactive poster presents recent results from research into an online professional learning experience for those new to teaching mathematics courses for future elementary school teachers. Course participants included graduate student, contingent, and tenure-track instructors from 2- and 4-year colleges. Results use three data sources: participant contributions in asynchronous discussion boards, verbal and textual communication during synchronous live sessions, and individual interviews. Questions driving the research: What did instructor-learners find challenging? Reassuring? How did they use what they learned? Why that way?
2020
Boston, Massachusetts
This report describes the first year of a pilot study to replace a non-credit bearing remedial algebra course with a credit bearing mathematics course designed for elementary education majors. The replacement course focuses on developing a deeper understanding of numbers and operations, compared to a primarily skills based remedial class. The resulting course average and passing rates of both courses, as well as passing rate of subsequent courses are included.
2020
Boston, Massachusetts
This study investigated students’ Quantitative Reasoning (QR) in STEM and Non-STEM math pathways using the Qualitative Literacy & Reasoning Assessment (QLRA). Participants were students who were enrolled in at least one college-level math pathway course at a large public institution in southeastern U.S. The results showed that STEM students scored, on average, higher than Non-STEM students. Both STEM and Non-STEM students who were further along in their math sequence had higher QLRA scores than those taking the gateway math courses in that pathway. However, students overall had relatively low QLRA scores, with an average score of 24%. These results indicate there is still a great deal of improvement to be made in students’ QR skills and that the math pathways initiative to align math curriculum with career fields and degree majors, while very important, does not invariably address quantitative reasoning.
2020
Boston, Massachusetts
Drawing on the commognitive framework, I studied student routines in the context of ring theory. Four micro-routine clusters emerged, as well as several macro-routines. Furthermore, questions about the commognitive framework and its operationalization were raised.
2020
Boston, Massachusetts
As part of a larger study, we looked at how Black women make sense of discouraging instructional events in their mathematics classrooms and future participation in the courses.
2020
Boston, Massachusetts
At this poster, we will discuss results of a teaching experiment investigating productive and problematic ways students assimilating with two levels of units reason about integration.
2020
Boston, Massachusetts
We report on the second iteration of the game Vector Unknown, a linear algebra game based upon the Magic Carpet Ride sequence from the Inquiry Oriented Linear Algebra (IOLA). Observations from the first round of gameplay interviews produced a variety of student strategies. These student strategies informed revising the game to introduce three different difficulties and the addition of a tutorial mode. In particular, we found that the presence of standard basis vectors could affect the strategies employed by the participants.
2020
Boston, Massachusetts
This poster presents a project aimed at developing and validating short tests to assess students’ comprehension of eight proofs that are commonly studied in undergraduate real analysis courses. We describe the project’s method, illustrating a process of fine-tuning items through the analysis of undergraduate students’ responses and mathematicians’ evaluations of these items. We also discuss some of the difficulties we encountered when designing proof comprehension tests in this context. In particular, we discuss challenges regarding (1) the heterogeneity of course curricular approaches to present these theorems and their proofs, and (2) the assessment of different facets of understanding a proof in this setting.
2020
Boston, Massachusetts
In this poster I explore the relationship between university mathematics professors’ meaning for average rate of change (AROC) and how they interpret student written work. Five professors’ meanings for AROC were characterized using two Mathematical Meanings for Teaching Secondary Mathematics (MMTsm) items. Their meanings were found to be robust, but difficult to connect to the different ways they interpreted student meaning in written work. I hypothesize that this difficulty is, at least in part, due to limitations in extending the MMTsm framework to characterization of post-secondary instructors’ meanings.
2020
Boston, Massachusetts
Many online materials are available to students learning calculus (e.g., Kahn Academy, S.O.S. Math, MIT OpenCourseWare, and Paul’s Online Math Notes) which were designed to help students in their courses, but not necessarily to help students develop conceptual meaning. We are designing an online environment on a website platform with the intent of meeting students where they are in their courses while also attending to recent advances in mathematics education research. We propose a study designed to investigate the meaning-making of students interacting with these online materials. In particular, this context provides windows into student thinking that are different from traditional written- or interview-based data sources. We seek input from RUME researchers on the design of studies related to the concept of concavity.
2020
Boston, Massachusetts
This study investigates the various knowledge types possessed by mathematics teachers in the teaching of statistics with the use of questionnaire as an instrument to measure teachers’ knowledge in the teaching of this discipline. This paper discusses the entire results on the knowledge possessed by prospective and in-service mathematics teachers in the teaching of statistics. The results revealed that there was no significant difference between the knowledge possessed by both prospective and in-service teachers in the teaching of statistics.
2020
Boston, Massachusetts
Mathematics education researchers have developed frameworks characterizing covariational reasoning. Here we present an early draft of an analogous framework for the covariational reasoning required in introductory-level physics. The framework is based on the premise that a proceptual understanding of physics quantities and their representations is the foundation of productive covariational reasoning in physics; physicists not only “imagine a continuum of input values in the domain of the function producing a continuum of output values,” (Carlson, Oehrtman, & Engelke, 2010) but also conceive and make sense of the inputs and outputs of each function as physical quantities. Our work indicates that physics experts use a number of distinct strategies to reduce cognitive demands so that covariational reasoning and making sense of physical quantities can be performed simultaneously. These strategies form the connection between sensemaking about quantity and successful covariational reasoning in physics contexts at the introductory level and beyond.
2020
Boston, Massachusetts
One expected student outcome of physics instruction is a set of quantitative reasoning skills that include evaluation of problem solutions. As part of a larger project, we developed and administered tasks to physics students that probe their use of these kinds of evaluation strategies. In a pair interview setting, we asked first-year students and juniors to evaluate expressions for the final velocities of two skaters involved in a one-dimensional elastic collision. The techniques used by the two groups show the differences between novice and intermediate versions of certain evaluation strategies. To do this, we focus on the role of algebra and mathematical operations in the checking process, how the students seem to view equations, and the different ways numbers are plugged into the given equations. By presenting this to the RUME community, we hope to gain insight into relevant RUME frameworks.
2020
Boston, Massachusetts
As part of an effort to examine student understanding and use of mathematical representations in quantum mechanics, three students were interviewed. In one task, developed to investigate student understanding of representations for probability concepts, students generated expressions in Dirac notation consistent with quantum mechanical formalism. While two of the students provided reasoning consistent with Dirac notation’s emphasis on vector concepts, the third did not appear to reason in this way at all. We use the symbolic forms framework to both analyze their reasoning and propose preliminary symbolic forms for Dirac notation elements.
2020
Boston, Massachusetts
No Abstract Listed
2020
Boston, Massachusetts
In this poster, we share results from a qualitative study investigating the instructional quality in algebra lessons at community colleges. Evidence and findings were gathered from a corpus of video data from fall 2017. We will present two codes, Instructors Making Sense of Mathematics and Supporting Procedural Flexibility, from our video analysis protocol. We will use these two codes to illustrate the interaction between instructor and content within the context of community college algebra instruction. We will explain the challenges in coding, implications for teaching algebra, and what we learned from our coding and calibration of videos.
2020
Boston, Massachusetts
The study reported here applies the principles of didactical engineering to design the teaching of mathematical induction. Three iterations of the process of didactical engineering have been designed and implemented with undergraduate students enrolled in a Discrete Mathematics course taught by one of the authors. Students’ performance on induction tasks improved with each iteration.
2020
Boston, Massachusetts
Building from Panorkou’s (2017) learning trajectory for dynamic measurement developed from elementary students’ reasoning with dynamic shapes, I use the results of a semester-long teaching experiment to demonstrate how covariational reasoning with a dynamic rectangular area context can extend beyond the elementary school classrooms to develop reasoning about rates of change as it relates to constructing formulas that are representative of the an equation resulting from implicit differentiation. Specifically, I relate a secondary mathematics pre-service teacher’s reasoning with dynamic area contexts to the learning trajectory proposed by Panorkou. I then identify her additional covariational reasoning used to construct formulas that re-presented relationships between the lengths and areas of the dynamic shapes. I conclude by providing suggestions for how her reasoning can be used to build towards meanings about implicit differentiation.
2020
Boston, Massachusetts
This poster serves to elicit discussion within the RUME community on experiences and observations the author has had submitting research conducted in lower-level mathematics classes. Analysis of past RUME proceedings shows that the RUME community focuses the majority of its attention in calculus and upper-level courses. Thus, the majority of undergraduates, in particular marginalized students, are not represented in RUME research.
2020
Boston, Massachusetts
To address a deteriorating classroom climate at the midpoint of a two-semester upper-division mathematics course sequence, we employed a novel instructor-led intervention: reading a mathematics education manuscript together with students as an invitation to legitimate peripheral participation in scholarly reflection on teaching and learning. This intervention resolved student complaints and promoted the idea of shared responsibility. We propose that reading mathematics education literature with students can be an effective tool for improving the climate of the classroom, and that using the didactical contract in this way can particularly help students claim their share of responsibility for their own learning.