RUME 26
2024
Omaha, Nebraska
Number of pages: 1494
Number of papers: 248
- Samuel Cook
- Brian Katz
- Deborah Moore-Russo
2024
Omaha, Nebraska
Number of pages: 1494
Number of papers: 248
This report (a) documents the phenomenon that student learning from online homework may occur across a series of mathematical tasks and (b) provides a theoretical explanation for the phenomenon by characterizing some of the cognitive mechanisms that underpin it. Specifically, findings indicate one way students learn from online homework is by comparing similar tasks. The results add to knowledge about what and how students learn from homework. They also have implications both for practice and research in terms of how we define and measure learning. In particular, the results support the need to consider and design for learning across multiple mathematical tasks.
While sameness is a theme that appears throughout mathematics courses, limited work has examined how multiple types of sameness are understood within the same course. In this paper, we examine survey responses from 49 discrete mathematics students who characterized how they would explain equivalence relations, numerical congruence, and graph isomorphism to a child. Results include language conveying notions of sameness in response to all three prompts but variations in how the sameness was framed for the different concepts. Implications include the need for more work characterizing nuanced differences in students’ understandings of similar concepts within specific courses.
Group isomorphism and homomorphism are core concepts in abstract algebra, but limited work has directly examined student conceptions of homomorphism or how students approach finding particular mappings. Based on interviews with two students, we contrast one student who used predominantly syntactic proof approaches for homomorphism with one who used semantic approaches, noting that both experienced success in finding homomorphisms in most cases.
Prior work has created approaches to calculus based on crucial quantitative reasoning. For integration, however, the major topic of u-substitution has generally not been fully detailed in these paradigms. This paper presents a study where students were taught u-substitution from a quantitative perspective based on a three-part quantitative structure: differential quantity, integrand quantity, bounds quantity. The students reasoned about the quantitative conversions in flexible ways, and used various quantitative relationship types in their reasoning. However, reasoning about the differential quantity was difficult, and a new type of “collapse” metaphor was identified. By the end, the students had all developed a good quantitative basis for u-sub.
Integration plays a significant role in applied problems, including those set in a wide range of physical phenomena. Past research has shown that students often experience difficulties applying calculus knowledge to physics, engineering, and other subjects. More specifically, recent research has identified students’ difficulties constructing the “product layer,” 𝑓(𝑥) ∙ 𝑑𝑥, as a core part of the definite integral. In this study, we extend research on students’ understandings of the definite integral by focusing on how they reason about multiplication with quantities, physical units, and the construction of the “product layer.”
Students’ understanding of the mathematics of change has been a crucial topic in the teaching and learning of precalculus. Various studies have indicated that covariational reasoning – reasoning about two quantities changing together—is critical for constructing productive meanings for rate of change. The Bottle Problem is one way teachers have supported students’ covariational reasoning. We report on a case study of one student – Kala – an Applied Precalculus student. Through analysis of her work on the Precalculus Content Assessment (PCA), course assignments, and a cognitive interview, we report on her reasoning with amounts of change—a level of covariational reasoning. As a result, we highlight the importance of using appropriate amounts of change to analyze situations covariationally. This study contributes to literature on students’ development of productive amounts of change reasoning.
Racialized gatekeeping in calculus courses is a national, systemic concern. However, research on equitable calculus instruction has focused on classroom-level changes, providing limited guidance for department-level reforms necessary to implement change in large calculus programs. To extend such work, we present a case study of a mathematics department at a large university engaged in improving its calculus program to better serve Black and Latin* students. Informed by critical whiteness studies, we explore how whiteness was maintained amidst this equity-oriented initiative. Findings exhibit disciplinary-specific forms of whiteness that stymied departmental reform. Implications are provided for equity-oriented, department-level change.
Incongruous mathematics self-efficacy (IMSE) is when a student’s self-efficacy does not match their performance. We present a subset of results from a larger qualitative study examining IMSE in collegiate intermediate algebra. We used qualitative interviews and surveys to longitudinally follow students for their entire semester in intermediate algebra and present the cases of three students with IMSE (over confidence). We show that participants’ over confidence seemed to stem from them doubting their performance would be predictive of their future success; participants did eventually lower their self-efficacy in response to repeated low performance. Results have implications for college mathematics instructors and for the study of self-efficacy.
Research on students’ construction and interpretation of integrals has grown significantly over the past decade. Some work has identified classes of interpretation for definite integral notation that include areas under curves, antiderivatives, and adding up pieces. The present study contributes to recent research on challenges students experience with the “product layer,” !(#) ∙ &#, when reasoning about adding up pieces. In particular, we present a case study in which 1 college student in a first-semester, calculus-based physics course drew on multiple knowledge resources associated with multiplication as she tried to construct a definite integral in a situation that for her was novel. Our results suggest that students’ understandings of multiplication with quantities is understudied in the literature on integration and an important direction for further research.
The relationship between mathematical induction (MI) and recursion compels us to ask how we could leverage recursive functions to bolster students’ understanding of MI. We describe task- based interviews that utilized concurrent interactions with MI tasks and recursive functions that mirrored those induction tasks via a character-based user-interface. To gain insights into how students’ conceptions of MI and recursion co-evolved as they interacted with these tasks, it was necessary to accommodate these multiple concurrent contexts by extending Hazzan's (1999) Reducing Abstraction framework. Our extended framework, called the Navigating Abstraction framework, documents ascending, descending, and transferring abstraction levels across contexts. Viewing the data through this lens allowed us to illustrate the way in which these three mechanisms together play a crucial role in students joint development of their understandings of both recursion and induction.
The practice of generalization is a fundamental aspect of mathematics that requires further investigation, particularly in advanced contexts. This paper has sought to address the research gap by presenting a preliminary framework that captures how and on what basis undergraduate students develop and evaluate mathematical generalization. I illustrate each aspect of the framework using examples from two focal students’ generalization of the concept of a group in Abstract Algebra, uncovering important mechanisms and thinking behind their emergent generalizing ideas. The framework has potential to be used more broadly in other mathematical contexts and to inform instructional designs aimed at promoting students’ generalization skills. Key takeaways and potential directions for future research are discussed.
Mathematics is central to STEM coursework, yet limited research has examined how instructors in math and other STEM courses facilitate connections between courses for their students. Based on four focus group meetings with ten faculty members, we characterize factors that assist or hinder making connections between courses and examine their relationship to professional obligations. Results include the largely negative impact of institutional obligations on faculty’s willingness and ability to make connections. Implications include the need for institutions to give opportunities to make personal connections across departments and incentivize the development of connections.
In this paper, I report on the findings of two students engaged in a teaching experiment on instantaneous rate of change (derivatives). While derivatives are a quantification of how two quantities covary, many students tend to have static images of the derivative (Zandieh, 2000). This teaching experiment was designed to support the students in building a meaning for instantaneous rate of change based in quantitative and covariational reasoning. The results of this study provide empirical evidence of the benefit of alternative teaching methods to the current standard curriculum.
This study explores undergraduate Calculus students’ perspectives on the Course Assistants (CA) program at a southern US university, where upper-class undergraduates guide weekly Math Lab sessions of collaborative problem-solving. Using sociocultural theories and qualitative methods the research investigates what aspects of Math Lab students and CAs found useful or not. Students' interviews and CAs’ weekly reflections indicate that, on the one hand, students by and large appreciate the experiences of collaborating over the weekly problems with their peers and the CAs, the social networking it provided, and the mathematical confidence it helped build. In our own words, students appreciated how the academic and social dimensions of math learning came together. However, many students and CAs are skeptical about the connections between the weekly tasks and success in the course exams. Relatedly, students often complained about Math Lab being another structured hour in an already time-consuming course and about varying student engagement.
This contributed report describes the development and validation of a new measure—the College Mathematics Beliefs and Belonging (CMBB) survey. The CMBB provides a contemporary measurement of undergraduate students’ perceptions of their mathematical proficiency and reasoning, beliefs about mathematics, and sense of belonging in mathematics. Primarily first- and second-year undergraduate students in five courses at a large public university in the United States completed multiple surveys to provide the data used for survey development. Confirmatory factor analysis (N = 935) and a reliability analysis indicate that the CMBB is a survey with fifteen factors that adequately measure various aspects of perceived mathematical proficiency and reasoning, beliefs, and sense of belonging. The CMBB survey is intended for use by both researchers and instructors to assess undergraduate students’ perceptions across these three domains with the aim of improving students’ experiences in college mathematics courses.
In this study, I describe how seven U.S. college calculus instructors framed instructional tasks for introducing derivatives symbolically to students. During one-on-one interviews, the instructors were presented with before and after student conceptions and were asked to propose tasks for introducing derivatives symbolically using the limit definition of derivative, both at a point and as a function. The instructors’ task framings reveal the types of mathematical problems students are expected to work on with respect to the content at stake: symbolic definitions of derivatives. Although the instructors heavily relied on calculating situations to introduce derivatives symbolically, some also used graphing, exploring, installing, and proving situations, which shows the high variability of task framing in calculus even when student conceptions before and after working on the tasks are predetermined.
The goal of this research is to investigate ways a dynamic geometry environment (DGE) and interactions during cooperative learning can leverage student understanding in geometry. Data from 18 students enrolled in a College Geometry course were collected by video recording in- class group work while students explored concepts in Taxicab geometry in a DGE. The textbook from this course and its activities are based on Action-Process-Object-Schema (APOS) Theory. As such, APOS Theory was used as a theoretical framework to analyze student reasoning during these activities. For this report, results are presented for one group of students and their discussion while working on an activity which encouraged the exploration of the mathematical definition of a circle in Taxicab geometry in a DGE. Trends emerged about how group structure while working in DGEs may influence interactions and outcomes for students. Some pedagogical suggestions are provided based on the results of this study.
In this report, we examine the use of grounding metaphors across three abstract algebra instructors in their discussion of quotient group instruction. We identified three primary categories: construction metaphors, equivalence metaphors, and positional metaphors. The construct metaphors were further refined into building metaphors and demolition metaphors. In this paper, we provide an overview of the types of grounding metaphors in usage and provide insight into how the different instructors took on these metaphors when describing quotient groups and their instruction. We conclude with some considerations as to how these metaphors provide insight into quotient structure and considerations for future research.
We report preliminary results of selected questions from a national survey of instructors of geometry courses for secondary teachers about the nature of instructor-student interactions. Survey responses (n= 118) are used to indicate six latent constructs describing aspects of instructor-student interaction that in turn quantify hypothesized characteristics of two didactical contracts, which we call inquiry in geometry and study of geometry. We found that instructors whose highest degree is in mathematics education are less likely to rely on a study of geometry contract than instructors whose highest degree is in mathematics. Also, instructors who have previously taught high school geometry are less likely to lecture.
Students learn about function composition, (𝑔 ∘ 𝑓)(𝑥), in secondary school. From two given equations, one might identify the composite function algebraically via substitution, 𝑔(𝑓(𝑥)). But what about when functions are given as graphs? This study aims to explore how students reasoned graphically with their revised responses to function composition tasks. We previously identified common types of resultant graphs participants generated in trying to sketch – in a short time period – the composition of various graphically-depicted functions. This paper specifically examines the ways in which students revised their graphs, when given more time to do so and after having engaged in trying to compose functions in a wider variety of contexts, and the reasoning that they provided for these changes. As such, we extend prior work by exploring how students reason about function composition when provided unlimited time constraints on their activity.
Undergraduate math education is highly inequitable. One potential strategy to improve the unfair experiences and outcomes of historically marginalized groups in mathematics is to use culturally sustaining practices, which relate math content to student’s culture and everyday lives. However, there is limited work exploring the applicability of culturally sustaining practices in the undergraduate mathematics setting. This study interviews seven math undergraduate instructors to better understand their perceived barriers to implementing culturally sustaining practices.
This article examines the relations between content, resource use, and the pedagogy of instructors in multivariable calculus. Content included topics that typically appear in multivariable calculus which were grouped into these general areas: introductory ideas, differential calculus, integral calculus, and vector calculus. The resources used included digital resources, 3D-printed models, and other physical resources. Pedagogy practices included both collaborative and individual in-class learning activities, instructor demonstrations, and homework. We also consider the instructor’s assessment of the activities they use. Data were obtained from three instructors working at three different types of institutions in the United States: a community college, a small selective undergraduate college, and a large comprehensive research university. The results provide information about instructional needs, as instructors implement resources in multivariable calculus, and about patterns related to the content they cover, the technology they use, and their pedagogical practices.
Of the many challenges in shifting towards inquiry-oriented instruction from a lecture-based approach is understanding the role of the teacher in this new paradigm. One aspect of that role involves navigating novel classroom authority dynamics as students bear more authority to create and justify mathematical ideas. Supporting teachers as they navigate this shift is one of the many roles of effective professional development (PD). This study of one PD facilitator’s authority as he worked to support instructors’ inquiry-oriented instruction (IOI) revealed a new type of authority, called pedagogical expertise authority, that may be of particular importance to understanding how to best support IOI. Additionally, the results of this study suggest the impacts of facilitators’ beliefs and teaching experiences on long-term professional development.
Postsecondary instructors interested in inquiry-oriented instruction of linear algebra participated in a sequence of eight one-hour online work group meetings with other inquiry- oriented linear algebra instructors and facilitators. Recordings were analyzed for how two participants referenced goals for instruction in discussions of implementing a new instructional unit on subspaces. We identified four goals for the instruction of teaching subspaces. We discuss the intersections of several goals that exist due to the tension caused by real-world contexts and abstract mathematical concepts. The instructors presented resolutions to the tension by utilizing varying teaching knowledge. Based on the results, we make suggestions for those who want to transition to inquiry-oriented instructional approaches.
Scholars and practitioners in higher education recognize that transformational change of organizations—especially departments and institutions—is difficult but essential to achieve needed, national-scale improvements in access, quality and equity in STEM instruction and career development. Based on studies of change projects in college mathematics education and gender equity on STEM faculties, we identify and describe a suite of common leadership approaches among change agents who led these projects. We propose that these approaches function as constructs for an emerging framework about change leadership. By observing how change agents lead complex change projects in higher education, we seek to develop theory about leadership for organizational change and to offer practical guidance to such leaders.
The use of inclusive teaching practices is novel for many instructors; there is a need to support instructors in envisioning what it looks like to teach inclusively. Related to this learning is a focus on reflection—on one’s current self as an instructor and envisioning whom one could become as an inclusive teacher. This study explores reflections made possible by reading and responding to collective transcription poetry. Informed by Gutiérrez’s conceptualization of equitable teaching as existing across critical and dominant axes, we took undergraduate mathematics program stakeholders’ definitions of inclusive teaching practice and created collective transcription poems. We presented the poems to stakeholders, who then reflected on them. We highlight reflections indicative of stakeholders’ current selves and possible selves, and the emergent theme of evaluative selves as ways in which to bridge these two dimensions of self with regard to inclusive teaching. We conclude by sharing directions for future work.
Despite existing research describing students’ understanding of group concepts, little research in undergraduate mathematics education has attended to students’ understanding of ring concepts. In this paper, we present an analysis of four students engaged in task-based interviews that provided insights into their understanding of quotient ring when constructed as an analogy to quotient group. Using university structure sense (Novotná & Hoch, 2008) as a foundation, we propose an expansion of structure sense to include attention to various structures (e.g., subgroups and quotient groups), and attention to analogous structure across different contexts (e.g., attending to quotient structure across group theory and ring theory.) Findings suggest that while students uniformly attended to similar structural components when creating the concept of quotient ring, there was variation in the depth of their reasoning about why certain structures exist.
This research design uses Approximations of Practice (AoPs), simulated practices of responsive pedagogy for prospective teachers to respond to, which lead them to make relevant connections between advanced mathematics courses and teaching practices. Data were collected from students in a Master's level Mathematics for Teachers course. The curriculum and interviews focused on three mathematical content domains: probability, algebra, and analysis. The AoPs were written into a semi-structured interview to demonstrate real-world examples of how teachers may see this material arise in the classroom. The AoPs prompted the participants to interpret and respond to student thinking and structure whole class discussions about students’ strategies. Their responses were analyzed using Wasserman’s (2022) pedagogical mathematical practices (PMPs). We examined each response to the AoPs to identify the participants’ PMPs.
Equivalence is a foundational idea in mathematics and a key fixture in the K-16 curriculum. There is considerable evidence, however, that students at all levels experience difficulties with it. A prevailing explanation is that students rely too much on transformations; and yet, transformational activity is absolutely essential: it is the primary means by which one generates more tractable representations that are better suited to the situation at hand. Strikingly, we found no studies that directly examine students’ productive uses of transformational activity. To this end, we conducted a series of task-based interviews with undergraduate students in order to illustrate and account for productive instances of transformational activity across undergraduate mathematics. Our findings affirm a hypothesis from the literature that supplementing one’s transformational activity with notions of equivalence can support productive reasoning. Additionally, we extend this idea by providing detailed analyses of what these supplementary notions of equivalence entail.
To better understand how students’ example and set use might evolve during their conjecturing and proving activity, we engaged two students in guided reinvention of mathematical statements relating sequence properties during an 11-week teaching experiment. We characterized the students’ evolving example/set use in three categories: (1) classifying examples from the initial set of sequences, (2) seeking diversity and using lack of examples from an expanded initial set of sequences, and (3) attending to properties, searching for structure, and building formality with the set of all sequences. We exemplify these categories and then discuss some guidance that could explain the students' example/set use.
This study builds on Louie, Adiredja, and Jessup’s (2021) sociopolitical turn on teacher noticing. In this study, I use graduate student instructors’ experiences as students and perceptions of their desirable actions to add nuance to the way Louie and colleagues discuss (anti-)deficit frames and noticing. The study uses a novel analytic framework to organize aspects of frames that begin to hint at a complex relationship between deficit and anti-deficit framing and responding.
Activities related to reading and understanding mathematical proofs are notoriously challenging for college students. On top of applying advanced logico-mathematical knowledge, these activities require the reader to process and operate on a substantial amount of information. Failure to successfully navigate the incoming stream of information may result in cognitive overload and impede one’s ability to make progress on the task. This case study investigates how an undergraduate student, David, used his hands to overcome the cognitive load he experienced when reading a proof of the Two-Color Theorem. The findings suggest that in the absence of other modes of offloading (e.g., using pen and paper, figurative materials, or technology), hand gesturing may serve as a powerful and convenient offloading mechanism.
We identify a current pragmatic and methodological problem facing RUME researchers who study processes that unfold over time, like classrooms and interviews, with qualitative coding techniques. In many cases, existing methods for estimating agreement between raters fail to account for the properties of this kind of data, are not compatible with how the coding schemes are applied, and fail to properly estimate rater agreement. Despite many peer-reviewers requesting estimates of agreement for coded data, there is often no suitable value to report and researchers must make do with claims about coming to consensus. We review methods found in the literature and evaluate their suitability, strengths, and weaknesses. While each reviewed method is appropriate for some aspect of this kind of data, none satisfies all desired criteria.
Understanding how students reason about the negation of logical statements is essential in supporting their mathematical development. Prior literature suggests that undergraduate students experience difficulties with the precise negation of logical implications. Many respond with the opposite statement “P implies not Q” when asked for the negation of “P implies Q.” We investigate the challenges that introductory proofs’ students experience when negating implications, and the reasoning they demonstrate in addressing these challenges. Our findings indicate quantification contributes significantly to their difficulties with negation. Further, nuances in students’ quantification may validate an implication’s opposite as its negation.
We present preliminary findings regarding the dimensionality of a 34-item instrument designed to measure the mathematical knowledge used in teaching college algebra at community colleges and the performance of items within the instrument. The instrument assumed a six-dimension model with two organizers of knowledge, one related to knowledge needed to perform two specific tasks of teaching, choosing problems and understanding student work, and the other related to knowledge of three function types, linear, exponential, and rational functions. Multidimensional item response theory models were applied to a sample of 416 community college mathematics instructors. A three-dimension model structured by function types better fitted the data than a unidimensional model. Two- and six-dimensional models, structured by the tasks of teaching or the combination of function types and tasks of teaching, did not converge so they are not discussed. We discuss implications and work to further validate the instrument.
Informed by Realistic Mathematics Education, we designed a hypothetical learning trajectory on graduate students’ guided reinvention of reducible and irreducible elements in unique factor- ization domains. We created experientially real tasks for use in a teaching experiment, in which students used algebra tiles as an emergent model of factoring integers and quadratics in ℤ[#]. In students’ mathematical activity, this became a model for abstracting the shared structure of (ir)reducible elements in ℤ and ℤ[#], which students used as they formally defined (ir)reducibles.
There is a need for mathematics instructors in higher education to have more equitable teaching practices. One way to address this need is through equity-centered professional development (PD). Using interviews across one year of professional development (PD), we describe how three immigrant mathematics instructors grapple with incorporating equity in their teaching practice using Gutiérrez (2009) four dimensions of equity. We found that the instructors focused primarily on awareness and access with respect to equity but did not understand how to infuse equity in their practice. We also found that three themes, the discipline of mathematics, all students are human, and the identities of the instructors and beliefs seem to serve as barriers for instructors’ implementation of equitable practices. Future research might consider how PD could be informed to account for the themes found in this study.
In this paper, we explore how undergraduate students use and evaluate generative artificial intelligence (genAI) in proving. We view proving as a human activity and proof as a production of proving, and hence, we believe that students need to be the ones who evaluate the genAI- generated arguments and write their proofs. In the initial phase of research on genAI in proving, we conducted interviews with three undergraduate students to examine how they use genAI in proving and how they write their proofs after their use of genAI. This study revealed that there is a flow in students’ use of ChatGPT in proving, and there are factors that impact their use of genAI. We think that the flowchart presented here can serve as a guide for both teaching and research on the use of genAI in the context of proving.
In this theoretical paper we leverage constructivist theories of learning to operationalize what is being learned during modeling. We do this by examining model construction through two theories: (i) schemes and concepts and (ii) abstracted quantitative structure. In addition to operationalizing what is being learned, we illustrate our operationalizations by providing examples from STEM undergraduates’ model construction activities.
Commutativity and associativity are recurring properties in school and university mathematics. In design experiments with pairs of secondary pre-service teachers (PSTs) we investigate what they are focusing when discussing commutativity and associativity, and how they deal with the notion of order. A qualitative discourse analysis shows for commutativity that the focus is on the order of two elements for most PSTs. Few focus on the order of several elements. Regarding associativity, bracketing is a main focus for many PSTs. The order in which operations are computed is sometimes discussed explicitly with also emphasizing the order of elements and sometimes more implicitly. The PSTs struggle with the notion of order so that we suggest coping strategies like “demanding precision in language” and “using alternative terms for order”.
As part of an effort to examine student mathematical and physical reasoning in quantum mechanics, a paired, semi-structured, think-aloud interview was conducted. The students were asked to interpret a quantum mechanical operator expression in a functional (position) representation and to think about an analogous expression in Dirac notation. When the students ceased to make progress on this task, they were given a modeling task in which they constructed an eigenvalue equation for a different operator in quantum mechanics. After examining their newly constructed eigenvalue equation, students were able to determine more precisely the nature of the original expression given in a functional representation. These data were coded in accordance with mathematical modeling and mathematical sensemaking frameworks to examine the intersection of the two frameworks.
This paper describes design issues for a conditional inference task with mathematical content. The task will mirror those used in cognitive psychology to study inferences from everyday causal conditionals: its items will present a conditional premise (if A then B) and a categorical premise (A, not-A, B, or not-B) and ask participants to evaluate whether a conclusion (respectively, B, not-B, A, not-A) necessarily follows. To assemble items, we asked six mathematics education researchers with expertise in conceptual understanding to generate conditionals covering a range of mathematical topics. To mirror the structure of tasks with everyday causal content, we asked that these conditionals should vary in believability. In this paper, we analyze the content and phrasing of the submitted conditionals in order to assess their suitability for use in a conditional inference task, and describe our planned use of this task to investigate the relationship between logical reasoning and mathematical expertise.
Lack of racial diversity has been an ongoing issue in higher education. Recently, the Theory of Racialized Organizations has been used to help explain why, despite many calls for diversity, the demographics of higher education have not changed. Considering this framework, we seek to understand what aspects of the graduate school application process are viewed as barriers by minoritized students for applying. As part of a larger study of undergraduate student knowledge of the graduate school application process, we analyze 515 responses from undergraduate math majors using Mann-Whitney U tests to identify differences in what participants view as a barrier to apply to graduate school by race/ethnicity. We discuss two main results and recommend changes to graduate programs wishing to recruit more minoritized students.
This study delves into undergraduate students’ ability to self-monitor and organize their claims coherently to ensure logical consistency when evaluating mathematical statements and validating accompanying arguments. To assess the capacity of undergraduate students to maintain logical consistency in mathematical contexts, we designed an online instrument comprising twenty statement-argument pairs, each rooted in mathematical content. We administered this online instrument to 205 undergraduate students, encompassing various levels of proof experience, across three diverse U.S. higher education institutions, including two large public universities and one small liberal arts college. Our analysis reveals a substantial number of undergraduate students displayed logical inconsistencies. It also appears that students may exhibit different levels of logical inconsistencies when the accompanying argument is framed using proof by contradiction, as opposed to direct proof. This prevalence underscores a critical concern in mathematics education, particularly proof-oriented mathematics.
Student understanding of integration and key concepts from Single Variable Calculus and the role of infinitesimals was investigated. Analysis was done using a framework which highlights quantitative reasoning. Data showed rich understanding of integrals in multiple representations, with some gaps. Also revealed were connections between those concepts expressed over intervals in Algebra and expressed instantaneously in Calculus, and how conceptions of infinitesimals supported those connections. Implications for instruction are considered.
Student projects in a postsecondary Quantitative Reasoning (QR) course can encourage students to think deeply about connections between a real-world situation and the corresponding mathematical or statistical model. These projects can help students collaborate and improve their 21st-century skills, such as critical thinking and oral and written communication. This paper reports how 13 QR instructors across two studies at 11 public postsecondary institutions in Ohio implemented student projects. We analyzed the course syllabus for each participating instructor, conducted at least one semi-structured interview, and observed their teaching using six Instructional Quality Assessment rubrics. Data revealed substantial variation in the implementation of student projects, resulting in varying opportunities for student learning. Our findings indicate that student projects are a critical variable in Quantitative Reasoning.
During the transition from procedure-based mathematics courses to proof-based mathematics courses, many students struggle to understand the purpose of examples. A productive use of examples can help both the generation and understanding of proofs. Although recent advances in research have argued the importance of productive uses of examples, there is little research investigating the alignment of research with current practices of instructors or of how the instructors view research on example-use. Findings from this study indicate instructors are aligned with mathematics education research on differing levels of example-use. However, many instructors may be hesitant to see examples be used by their students in written work. Despite this hesitation, instructors still encourage students to use examples to aid their understanding of proof.
Problem posing is an important part of mathematical inquiry, but students can struggle to pose problems that are mathematically relevant. One route for supporting meaningful problem posing is through playful math tasks, which can emphasize agency and exploration. In this paper we report findings from a small-group teaching experiment with five pre-service secondary teachers who explored a variety of tasks, including open tasks, problem-solving tasks, and tasks designed to foster mathematical play. In investigating students’ problem posing, we found that developing playful challenges for one another supported instances of discovered complexity, the experience of new conceptual challenges. We discuss examples of discovered complexity and the mathematical ideas students developed when grappling with novel ideas.
Implicit differentiation is an important topic in first-semester Calculus, yet only recently have researchers in the RUME community turned their attention to it. Our study seeks to contribute to this growing body of knowledge. We examined students’ performance on an instructional activity that emphasized the graphical representation of implicit curves alongside the symbolic competence of implicit differentiation. The students worked in small groups during recitations and then completed a similar type of question on an assessment. A researcher-designed Hypothetical Learning Trajectory was used to structure the instructional task and to analyze student written data. The results suggest that students’ attainment of symbolic learning goals is higher than that of graphical ones, with most difficulties experienced with coordination between symbolic and graphic modalities. Implications for further research are discussed.
Even though algebraic conceptual understanding is recognized as a critical skill, existing larger- scale validated algebra assessments consist mostly of computational tasks, or only assess a very narrow range of conceptions in a smaller focused domain. Further, few instruments have been validated for use with college students. In this paper, we describe the creation and validation of an algebra concept inventory for college students. We describe how items were administered, revised, and tested for validity and reliability. Results suggest that algebraic conceptual understanding is a measurable construct, and that the instrument has reasonable validity and reliability. Revision and validation is ongoing; however, lessons learned thus far provide information about what conceptual understanding in algebra might look like and how it might be assessed.
In this contributed report, I tell a story surrounding one mathematician’s learning to teach mathematics content courses designed for prospective elementary teachers. Namely, I describe the mathematician’s background and the series of events that pushed this newer mathematics instructor to develop productive dispositions towards their students, leverage support from their mom, a former elementary teacher, and navigate challenging student questions in the classroom. This report draws on results from my dissertation study, a narrative inquiry of one mathematics instructor, involving a semester-long collection and analysis of classroom observations, observations of instructor meetings, weekly instructor reflection journal entries, three instructor interviews, and instructor-written mathematics teaching and learning autobiographies. A key implication is that mathematics instructors may benefit from early experiences developing productive mindsets towards teachers and teaching, and need access to individuals who value teachers and teaching and who avoid deficit discourses around teachers or students.
Here we explore how college students across a wide range of courses may conceptualize symbolic algebraic properties. We draw on the theory of Grundvorstellungen (GVs) to analyze how learner conceptions may or may not align with instructional goals. In analyzing interviews, several categories of conceptions (descriptive GVs) emerged that may help us to better understand how students conceptualize symbolic properties during instruction.
In this report, we provide an initial exploration into a key but under-studied phenomenon in enumerative combinatorics – the use of division in solving counting problems. We present a case of one undergraduate student solving a combinatorics problem; this case is representative of a broader phenomenon in which students may intuitively desire to account for an overcount using subtraction, when division is a productive and useful approach. We highlight the conceptions a student demonstrated as she progressed from using subtraction to using division successfully. We frame our analysis in terms of a set-oriented perspective (Lockwood, 2014).
University mathematics departments are making efforts to improve student success in STEM by implementing active learning in their introductory mathematics courses. These efforts are context-dependent, driving a need to understand the varying paths departments take in making these changes. In this paper we explore the change efforts of two mathematics departments that achieved varying levels of success. We take a longitudinal perspective to capture their efforts. We identify what levers these departments used, including how robustly the levers were implemented. We conclude with a discussion on how the levers identified contributed to each department’s progress.
This study explores different ways that linear algebra students reason with a non-traditional linear system, referred to as the Gulliver system, in a task-based clinical interview. Using the constructs of Naming and Locating developed in the conceptual framework, an a priori analysis outlines how students may engage in Locating and Naming tasks. The a priori analysis was used for data analysis as a basic framing. Students’ engagement with the non-traditional linear system and the refined and extended a priori analysis will be presented. Students’ adoption of their previous experience with the Cartesian coordinate system will be also discussed.
While the pandemic brought much hardship for mathematics education, it also brought opportunity for ingenuity and transformation. This study builds on results from prior work that found Google Docs to be rich tools that supported students in effectively engaging in collaborative proof activity in a remote introduction-to-proofs course. To study if Google Docs would have similar benefits for students’ collective proof activity in a face-to-face setting, we investigate how two students in a face-to-face course leveraged a shared Google Doc to support their collaborative work on a proof construction task. We found that the students used the shared Google Doc to support their collective proof construction in similar ways as students in the remote class. In addition, we observed that the Google Doc was an inherent component of students engaging in an active proof writing process.
Students enter undergraduate mathematics classrooms with a variety of mathematical backgrounds. We view these past mathematical experiences as assets that support students in their future mathematical learning. In this paper, we seek to characterize these assets by introducing a new concept in mathematics education: mathematics capital. This concept emerges from a framework connecting Archer et al.’s (2015) conceptualization of science capital and Bourdieu’s seminal work on the forms of capital (1986). Survey data were gathered in Spring 2023 from a total of 219 students in undergraduate mathematics courses at an urban midwestern university. Path analyses in structural equation modeling showed a strong association between the variables of mathematics capital and self-efficacy. Other associations are also discussed. Our results indicate that mathematics capital is a measurable, quantifiable variable, independent from others. Future research is ongoing to better understand the nature of mathematics capital and how it is related to other variables.
This paper explores the intersection of two critical areas in undergraduate mathematics education: the pursuit of diversity, equity, inclusion, and accessibility (DEIA) in mathematics classrooms and the understanding of equity by mathematics graduate teaching assistants (MGTAs). MGTAs play a pivotal role in teaching undergraduate mathematics courses, yet their grasp of equitable teaching practices remains underexplored. In this study, we investigate 21 MGTAs’ conceptions of equitable teaching. The results indicate the importance of considering the critical axis of Gutiérrez’s (2009) equity framework. Ultimately, the findings offer insights into MGTAs’ conceptions of equitable teaching, providing a foundation for developing effective professional development programs and advancing DEIA efforts in mathematics.
Studies have described a number of blockages to mathematizing, the process of transforming a real-world situation into a mathematical model. Recently, researchers have documented four cognitive obstacles associated with those blockages in physics-based modeling tasks. We attempted to extend these findings to biological-based modeling tasks. However, we found it difficult to use the theoretical lens as described in the previous literature. This difficulty arose because the previous theoretical lens required the delineation of real-world objects and mathematical objects, a pursuit recently shown to be unattainable in some contexts. By examining students mathematization under a quantitative reasoning and symbolic forms lens, we were able to find two cognitive obstacles analogous to the ones found in previous research, confirming their existence with a different demographic of students and different task scenarios.
Teaching professional development (TPD) in collegiate mathematics has expanded over the last few decades. Providers of TPD, people who organize and facilitate professional learning about teaching, are at the center of this growth. Yet, little is known about who Providers are and what they do. To better understand the national landscape of Providers of TPD within university mathematics departments, this report shares data from a national survey where respondents were Providers. The focus here is on findings from survey questions asking about characteristics of Providers and the “providees” with whom they work, along with formats, topics, and activities used in TPD. Results suggest that Providers value active, learner-centered instructional methods promoted by research and policy. However, in the TPD itself, formats, topics, and activities commonly used by Providers may preach but not regularly practice activity-based methods.
The ACT UP Math project is studying the role and impact of research-practice partnerships between mathematics education experts and mathematics department faculty to critically and systematically initiate transformative efforts to improve the experiences of students with marginalized identities in introductory mathematics programs. This paper explores the shared experiences we encountered while working toward critical transformations of three departments from within those departments, focusing on three tensions that surfaced and framing these tensions as expected and necessary components of the process of critical change. These three tensions explore the role of identity as neutral or central, the enactment of power as power over or power with, and the role of students as experts or novices. These expected and necessary tensions are evidence of the transformations from dominance toward criticality happening within the departments.
This study observed real analysis students attempting the homework assigned by their instructor to identify how they used resources, specifically example proofs, to complete their homework. The data consisted of seven students in total over three instructors. Overall, the study found that students tended to adopt a recurrent style of how they used their notes, or did not use their notes, to advance their solutions. However, there were some instances where the students would deviate from their style which could be linked to the type of problem as well as how the instructor taught that topic. This study contributes insight into how real analysis students may use their resources for the tasks assigned by the instructor.
It is essential to provide opportunities for prospective secondary mathematics teachers to connect advanced mathematics content to secondary mathematics teaching practice, if advanced mathematics courses are to be useful to these teachers. We argue that for teachers to frame teaching and learning in equitable ways, connections to teaching must represent learning in anti-deficit ways. We review a selection of curricular materials satisfying two criteria: first, they are written for use in advanced mathematics courses that prospective teachers may take; and second, they feature explicit connections to secondary teaching practice. We find that representations of learning in curricular materials tend to take binary views of mathematical legitimacy, and frame students as either being correct or incorrect. We conclude with implications for mathematics teacher educators in RUME.
This papers examines the stories two instructors tell about why formally defining real numbers in the context of a Real Analysis course is a desirable objective. While the two stories exhibit some variations, the analysis shows that both narratives relied on the following constitutive elements: (1) calculus is a desirable activity, (2) students’ current understanding of R is lacking, and (3) to use R for calculus, there is a need to specify it precisely. I discuss how these assumptions can be misrepresenting of practice and experienced as problematic impositions on students.
We surveyed university mathematics instructors across the United States about their planning practices. We sought to better understand the complexities involved in the work that goes into preparing for instruction at the university level. Research indicates that the quality of instructors’ lesson plans can be linked to the quality of their instruction (Akyuz et al., 2012). Instructors self-reported as either using inquiry-based practices, lecture-based practices, or an even mix of both. Instructors that indicated using both inquiry-based and lecture-based practices spend more time lesson planning, but often feel less supported by the lesson plans they create. These instructors attempt to do practices that both the inquiry-based instructors do (i.e., spend time preparing for and accounting for student thinking) and lecture-based instructors do (i.e., ensuring a clear understanding of the mathematics for themselves). We discuss implications for instructors and the undergraduate mathematics education community.
The VIP-Math project aimed to understand whether and how teaching practices of mathematics instructors are influenced by discussions of their classroom data visualized using the TAMI-OP classroom observation tool. We found that TAMI-OP data visualizations supported discussions that led the participating mathematics instructors to set both short- and long-term teaching goals, and to experiment with teaching practices related to at least one of their goals. Given the limitations of the participant pool, further research is recommended with a greater diversity of instructors and coaches.
Using Balacheff’s (2013) model of conceptions we analyzed textbook examples in the linear independence section of an interactive linear algebra textbook and 61 student responses to a similar reading question. Reading questions seek to entice students to read the textbook before attending the lesson when the ideas will be discussed; the responses are immediately available to instructors. We found ten additional parts of conceptions (control structures) used independently or in combination with the ones promoted in the textbook. We discuss the implications of our findings and our plans for future research.
Undergraduate teaching and learning assistants (UTLAs) can support learning and students’ sense of belonging in undergraduate, active learning STEM classrooms. One main goal of incorporating UTLAs into a classroom is to connect students to a peer-like instructional figure whom, it is assumed, students find more relatable. However, little is known about how UTLAs change the social dynamic of undergraduate mathematics classrooms, particularly those taught by an instructor of record who is a graduate student (GSI). I used positioning theory as a lens to understand how students locate UTLAs in instruction. I analyzed 411 undergraduate precalculus student responses to the survey question: “Do you interact differently with [the GSI] than [the UTLA]? If so, please explain.” From a qualitative analysis of responses, I found that students positioned UTLAs along three storylines that have different implications for the distribution of instructional duties between UTLAs and GSIs in an active learning environment.
This study investigates the characteristics of an instructor's communication in one-on-one oral assessments for an introduction-to-proof course and how they impact student learning. It examines the instructor's strategies for evaluating the depth of students’ knowledge. Using thematic analysis, the study examines patterns in question types and their relation to revision quality and presentation fluency. Findings reveal that well-revised proofs prompted questions about comprehension and logical structure. For revisions needing improvement, questions guided students to establish correct proofs before addressing comprehension. The study also explores teaching elements resulting from the instructor's assessment practice using commognitive theory. These findings contribute to assessment practices, particularly in revision and oral assessment for introduction-to-proof courses. By highlighting communication's role in student learning, this research advances alternative assessment understanding in higher education mathematics.
This study investigates undergraduate students’ meanings for quantified variables, and how the language of a mathematical statement impacts their meanings. I conducted 3-day exploratory teaching interviews with eight students from transition-to-proof and advanced calculus courses (four from each course). Over the course of the three days, I presented students with a variety of statements from calculus in order to test how both and language and mathematical content might impact students’ meanings for quantified variables. I found that students’ meanings did often change across mathematical contexts that contained different language. In particular, the words “any” and “all” were found to lead some students to use different meanings for quantified variables in calculus statements.
Studies on the use of technology in calculus courses have focused on the use of different types of technology as learning tools for visualization, representations, and understanding. However, research on students’ perspectives about the use of technology in calculus classrooms needs to be expanded. In this article, we interviewed eight calculus students in a US research institution to understand their perspectives about the use of technology. To analyze the data set, we used thematic analysis informed by the technology acceptance model focused on external variables related to human factors. As a result, students identified human factors that may influence their perspectives about using technology in calculus classes such as: their attitude and skills, selection of mathematical problems, feedback, classroom environment, and resources. These findings inform calculus instructors and coordinators about possible considerations before implementing the use of technology in their classrooms.
n this paper, we compare the types of teaching feedback that graduate student instructors pro- vide their peers in comparison to more senior faculty at a large research-oriented university. Additionally, we consider the challenges and benefits that graduate student instructors report concerning providing teaching feedback to a peer. Our results reveal that graduate student in- structors and faculty contribute distinct perspectives on teacher growth and together can form a strong support system for first-time graduate student instructors. Additionally, while observing a peer does pose real challenges, we found that graduate student instructors develop strategies to overcome these and report more benefits than difficulties.
We investigated teaching practices aiming to communicate multiple uses of a common term in mathematics with students focusing on the use of a single word “derivative” for both objects “the derivative at a point” and “the derivative function” using commognitive approach and intellectual needs. Our analysis of a teacher’s teaching explicitly attending to this feature revealed several teaching practices with this aim: discussing discursive rules for distinguishing the two uses and using another mathematical object for connecting the uses (e.g., slope); discussing a colloquial object that has a similar feature and is familiar to students, which provided a familiar discourse that students could rely on; and avoiding multiple uses of a common word in one context, Those teaching practices are impacted by multiple uses of a common signifier in communication and mathematical relations among the two objects objectified by a common word, but also impacted how the objects became related.
Proof is a central pillar of mathematical practice, and so teaching proof to undergraduate students is a necessary responsibility for undergraduate mathematics instructors. Our study contributes to research on how instructors can achieve this goal by examining their responses to student-generated existence proofs. In particular, the simple logical structure of an existence proof allows for proof-writing norms to be the focus of attention. By analyzing instructors’ responses to students’ existence proofs in terms of norms, values, and professional obligations, our study attends to the ways in which norms are (or are not) supported by instructors in undergraduate mathematics classrooms.
In this study, we report one group of students’ efforts to create a community meaning for set- builder notation collectively. Students’ ability to develop and interpret set-builder notation is essential to transition-to-proof courses. Conventionally, a colon is used in set-builder notation to (1) separate the universe of discourse from the set’s defining property and (2) indicate an ordering to these components, with the universe to the left and the property to the right of the colon. We describe one normative and non-normative interpretation of this notation and how the students’ individual attribution of conventional meanings for the colon to different inscriptions within the notation helped (or inhibited) them from interpreting these expressions. We report how communicative discourse between the students affected their meanings and discussions.
Serving diverse student populations equitably is a focal concern for mathematics educators (Association of Mathematics Teacher Educators, 2017), particularly given recent teacher shortages in high needs schools. Teacher preparation programs are tasked with preparing new teachers to thrive in these settings. In this paper, we examine what preservice, secondary mathematics teachers found valuable engaging in structured mentoring and guided reflective opportunities that integrate theory into practice. Participants engaged in authentic experiences including learning assistantships along with traditional practicum experiences. Participants completed guided written reflections throughout the semester, in addition to meeting regularly with a faculty mentor. We utilized Wenger-Trayner and Wenger-Trayner’s (2014) value framework to examine the data and share findings that suggest most participants developed an awareness of mathematical content knowledge, pedagogical content knowledge, and knowledge of students (Ball et al., 2008) at an earlier phase of their training than may be expected in traditional teacher preparation programs.
Reading didactical texts, such as textbooks, requires particular sets of literacy practices. Researchers have proposed that these practices are related to aspects of identity and agency, and have described aspects of these constructs separately. In this study, we examine the interactions between students’ agency and their macro- and micro-identities. We present data from two undergraduate students reading a section of a calculus textbook and explore the ways the interactions between the students’ identity and agency shape their reading activity.
Understanding mathematics identity development has yielded insights for undergraduate mathematics education. In this work, we build on existing conceptualizations of mathematics identity to explore this construct in the context of mathematics graduate students with academic career aspirations through analyses of “Dear Math…” letters. We identify and detail four dimensions of graduate students’ mathematics identity: Doer of mathematics, Sharer of mathematics, Feeler of mathematics, and Believer of mathematics. We conclude by discussing implications of these dimensions, as well as the unique opportunities of “Dear Math…” letters to provide valuable insight into graduate students’ mathematics identities.
This report discusses the validity evidence reported in undergraduate mathematics education measures. Through a comprehensive literature review of articles published from 2000-2019, we identified 166 measures that provided validity claims and evidence to varying degrees. Findings overall suggest that validity evidence can be more robust within measures of RUME constructs, although there are examples of rigorous validity arguments. Of the validity evidence, the major sources of validity were reliability, test content, and internal structure. We report on the number of sources of validity, by construct, and the types of evidence to support the validity claims.
This study investigates students' conceptions of what makes a particular written mathematical communication a proof. To that end, we implemented one-on-one task-based interviews involving mathematical communications. These communications were presented as trios, and we called each communication a job. We intentionally designed some of these jobs to sit in the vague space between proof and calculation. Our study investigated the reasons why participants believed certain jobs were proofs, allowing us to develop a model of students' conceptions of proofiness. This model revealed differences among students in terms of which attributes they thought contributed to or detracted from proofiness, as well as those they thought were and were not relevant to whether a job is a proof. These differences and how they relate to normative conceptions of proofiness point to aspects of proof that may require further attention in these students' introduction to proof courses.
While active learning has been found to support student success, it has not yet emerged as the dominant instructional approach in undergraduate mathematics (Stains et al., 2018). This study focuses on student performance in undergraduate Calculus I where we compared the performance of students in inquiry-based (IBL) Calculus I with students in non-IBL Calculus I courses. Additionally, we examined the performance of first-year and upper-year students as well as students who were or were not pursuing a math-intensive major. Students in IBL Calculus performed on the final exam an average of nine percentage points higher than those enrolled in non-IBL Calculus. Differences in performance between first-year students and upper-year students and between students majoring in math-intensive fields and those in non-math-intensive narrowed in the IBL classes. This study aims to add to the ever-growing body of research showcasing the benefits of IBL in improving student course performance.
Undergraduate mathematics classrooms are racialized spaces for Latin* students, even at Hispanic-Serving Institutions (HSIs) with educational missions of cultural affirmation. Instruction plays an important role in reinforcing and disrupting racial oppression in mathematics, which has significant implications for gateway courses (e.g., calculus) that impact STEM persistence. Groupwork is a widely-adopted practice in gateway mathematics courses with intentions to promote equitable access to content and participation; however, research has shown that groupwork can perpetuate inequitable experiences for historically marginalized groups in STEM, including Latin* students attending HSIs. The present study addresses these concerns of racial equity in undergraduate mathematics by exploring Latin* students’ groupwork experiences in gateway courses at a HSI. Our findings capture how groupwork facilitated or removed access to a sense of racially-affirming community, which was central in Latin* students’ visions of equitable support as mathematics learners at a HSI.
In mathematics, counter narratives can be used to fight the dominant narrative of who is good at mathematics and who can succeed in mathematics. Eight mathematicians were recruited to co- author a larger NSF project (RAMP). In part, they were asked to create author stories for an undergraduate audience. In this article, we use narrative analysis to present five polarities identified in the author stories. We present various quotations from the mathematicians’ author stories to highlight their experiences with home and school life, view of what mathematics is, experiences in growth in mathematics, with collaboration, and their feelings of community in mathematics. The telling of these experiences contributes towards rehumanizing mathematics and rewriting the narrative of who is good at and who can succeed in mathematics.
Active learning in STEM classrooms has been shown to increase student outcomes in multiple ways. We present here a discussion of data analysis supporting these types of conclusions using data from a large-scale randomized study of the implementation of an active learning-based curriculum for Calculus I that collected 1019 observations of student outcomes over three semesters. We compare fixed-effects models including cluster levels of student learning outcomes to mixed-effects models with random cluster level effects. We discuss the differences between the models and the resulting effect sizes suggested by the different factors included in the models.
The relevance of upper division mathematics courses for future secondary teachers is a longstanding thorny issue. Suggested improvements include capstone courses and revised upper division content courses to explicitly address future teachers’ relevant secondary mathematics content knowledge, beliefs about teaching and learning, and experience with learning mathematics while engaging in authentic mathematical practices. In this report, we investigate prospective teachers’ reflections on their opportunities in an upper division Inquiry-Oriented Dynamical Systems course to engage in the eight Common Core State Standards for Mathematical Practice. Analysis of students’ self-reported engagement in the eight Practices revealed five practices that strongly resonated with them and the various ways that their experiences in an inquiry-oriented classroom supported meaningful and powerful engagement in these Mathematical Practices. We conclude with implications for practice.
Efforts have been made to study (in)equity in undergraduate mathematics education research. Across various fields, there is a foundation of work on how status impacts students’ learning and participation in small groups as well as how differing patterns of interaction contribute to inequitable outcomes. This report contributes a networked theory for analyzing relationships between status and (in)equity in general. We argue that applying this methodology to proof-specific contexts has the potential to uncover how status hierarchies form in proof classrooms using group work components. We hope to promote conversations within the RUME community around taking actionable steps towards delegitimizing status hierarchies in proof classrooms.
Functions are critical in mathematics but have received limited attention at the advanced level. Research in advanced contexts has primarily focused on students' reasoning about specific types of functions (e.g., isomorphisms) but not on the function concept itself. In this paper, we explore students' productive techniques involving the definitive properties of function: well-defined and everywhere-defined. We found that the techniques students productively employed extended far beyond canonical procedures (like the vertical line test) and largely drew upon function meanings involving coordination of the domain, codomain, and rule, which previous research has highlighted the importance of but stopped short of directly investigating. Two contributions of this work include our focus on productive, successful techniques (rather than challenges and difficulties), and explicit focus on everywhere-defined (which has not received direct attention).
In this paper, I describe two vignettes of two students’ meanings for partial and directional derivatives. The data was collected in the Spring 2023 academic semester from two STEM students who had completed multivariable calculus at least two semesters prior to participating in the study. The interviews were conducted using task-based, exploratory teaching interviews with the goal of creating explanatory models of student thinking to highlight the nuance in their meanings for differential multivariable calculus ideas. The two students, Alonzo and John, each exhibited novel meanings for partial derivatives that were constructed over several semesters of mathematics and physics instruction. These results add to the field’s understanding of students’ thinking about a relatively underdeveloped area of the research literature, specifically multivariable calculus education.
Online mathematics tutoring has become a fixture of the educational landscape in higher education. Using Lepper and Woolverton’s (2002) INSPIRE model for effective in-person mathematics tutoring as a basis, we developed a codebook to analyze online tutoring, modifying existing codes to reflect how they manifest in the online environment and augmenting our codebook with behaviors that were not present in the in-person environment. We give a description of the current state of online mathematics tutoring and identify methods used by online tutors to actively engage students in the learning process. In addition to our results, we include a brief description of some challenges we faced as researchers when engaging with research on online tutoring and the solutions we reached.
Graduate Record Examination (GRE) scores are commonly required in applications to graduate school in mathematics. We examine undergraduate mathematics majors’ knowledge of the GRE and their perceptions of the GRE as a barrier to applying to these programs as part of a larger project studying student knowledge of the graduate school application process and how it contributes to lack of diversity in graduate mathematics programs. We found that there was an association by gender, and that women were less likely to report that they had heard of the GRE General and Subject Tests. Similarly, women were more likely to report that the GRE tests were a potential barrier to their decision to apply to graduate mathematics programs.
Introductory calculus classes often serve as prerequisite classes for science and engineering majors. However, some researchers have questioned whether calculus courses, as currently taught, are filtering students appropriately (Black & Hernandez-Martinez, 2016; Williams, 2012; Williams & Choudry, 2016). Central to introductory calculus classes is the derivative (Kidron, 2019). Scholars have discussed the ways in which representations of the derivative differ between disciplines (Dray et al., 2019), but there has been no formal research investigating the representations of the derivative valued by teachers of calculus and teachers of subsequent non-mathematics classes. This study, through a new survey instrument, determined there are significant differences between the representations of the derivative valued by various post-secondary teachers in majors requiring calculus. By analyzing these differences, recommendations can be made to improve calculus as a preparatory course for non-mathematics majors.
Undergraduate mathematics classrooms continue to experiment with active learning strategies as an alternative to the traditional lecture-based teaching model. This paper investigates one student’s (Jacob) engagement in a poster session activity in a Calculus 1 course and explores how he reasoned about calculus concepts in this non-standard learning environment. Informed by Sfard’s theory of commognition, an analysis of Jacob’s discursive routines reveals a complex interplay between ritualized and explorative discourse. While Jacob’s poster presentations appear highly ritualized on the surface, with routines recycled from class, indications of explorative activities emerged in his preparation with a partner, reflecting an active search for suitable routines. This research emphasizes the importance of considering the entire ritual- exploration continuum in mathematics education and raises questions about facilitating the transition from ritualized to explorative discourse.
This qualitative case study attempts to understand instructional goals that a mathematics instructor in India sets and envisions for engaging students. The instructor participated in a professional development program provided by the authors on the active role of students in learning mathematics. The authors’ interpretations from three interviews were guided by two theoretical lenses of Realistic Mathematics Education and Rehumanizing Mathematics. In return, the interpretations offer a context to imagine how the lenses might interact.
Mathematics education research has linked students’ development of positive mathematics identities to outcomes of academic success and persistence in STEM. I contribute to this research area by adopting a positional approach to identity development, focusing on the students’ development of in-the-moment identities through the negotiation of acts and roles. To better understand the connection between the roles adopted by students in class and the students’ mathematics identity development, I conducted classroom observations of an undergraduate precalculus classroom and interviewed four focal students on their beliefs about math and experiences in this course early and late in the semester. Student-instructor interactions from the observations were coded to produce positioning profiles for the focal students. Here, I present the positioning profile for one student, alongside a reflection of her transition from not identifying as a “math person” to identifying as one, as developed through narrative analysis of the interview data.
Within secondary mathematics teacher preparation, recent scholarship points to the efficacy of strong links between university mathematics course content and secondary mathematics teaching practice. Studies aligned with this view typically focus on overarching course design, rather than instructional characteristics. This investigation builds on a prior such study that found improvements in teachers’ potential competence for teaching across four different content areas. In the present study, we analyze 137 teachers’ expectation for success in and valuing of core instructional practices and their gains in content knowledge for teaching, as well as a purposive sample of videos from 6 courses enrolling a subset of these teachers. Based on our analysis, we suggest that different aspects of university instruction may have differential influence on attitudinal and cognitive aspects of teachers’ potential competence. Moreover, instructional practices that most benefit attitudinal aspects may be in tension with those that most benefit cognitive aspects.
We designed approximations of practice (AoPs; Grossman et al., 2009) that pose hypothetical classroom scenarios of high school students learning about factoring polynomials. We investigated the ways in which prospective and in-service teachers leverage their understanding of unique factorization domains to inform their teaching practices of attending, interpreting, and deciding how to respond to students’ thinking in the AoPs with the goal of making explicit connections between abstract algebra and the teaching of secondary mathematics.
The transition from undergraduate to postgraduate studies has been underexplored in mathematics education research. Several studies showed that this transition is challenging as it requires countless changes in familiar practices. In this paper, we present the case of Jordan— an honors student who was developing mathematical background as a stepping stone toward his research project. With the commognitive framework, we analyzed the data from three interviews to reveal Jordan’s mathematical learning network of routines. The findings detail the network’s components and illuminate the role of agency in Jordan’s network development. Agency appears as necessary to navigate learning routines common in undergraduate studies and enables Jordan to be less dependent on externally provided resources shaping his learning.
In this study, we conducted a teaching experiment to test hypothetical learning trajectories of reinventing the definition of the unique factorization domain. Six graduate mathematics students who are preservice and in-service teachers were given experientially real tasks of examining factorizations of integers and quadratics using algebra tiles. The experiment showed that their intuitive and informal reasonings about the reducibility and unique factorization were leveraged to formalize two defining axioms of the unique factorization domain by the emergent model of algebra tiles and by the pedagogical moves of the teacher-researcher who led the sessions.
This study explores the instructional practices of four graduate teaching assistants (GTAs) – Ursula, Ava, Phoenix, and Cole – in an active learning Calculus I course. Each GTA's unique pedagogical background is discussed in relation to their teaching methods and their students' classroom actions. Ursula, a seasoned high school math teacher, brings a wealth of experience in active teaching, while Ava, having encountered active learning as a student, reflects on her positive experiences. Phoenix, a novice GTA, emphasizes group work and individual student exploration, while Cole, with limited teaching and active learning experience, promotes student independent work. The contrasting approaches of these GTAs within the same purposefully designed active learning environment provides valuable insight into the connection between previous educational experiences and teaching practice, especially with respect to student engagement in a Calculus classroom.
The perspectives, habits, and mindsets of experienced instructors and coordinators serving within course coordination systems are valuable for effective coordination. In this paper, we present perspectives and recommendations from coordinators and instructors being coordinated with the goal of helping those who are new to coordination. We utilize Martinez et al.’s (2022) work on two coordinator orientations, Humanistic-Growth and Knowledge-Managerial, to frame the common perspectives of instructors and coordinators of first-year mathematics courses. Most participants gave reasonings for their decisions surrounding coordination with a healthy balance of Humanistic and Managerial considerations in mind.
A recent movement has developed to emphasize the importance of inclusive teaching practices in undergraduate STEM education. Mathematics courses have one of the largest impacts on students’ persistence and success in STEM majors, making efforts to improve diversity and inclusion in these classes all the more important. In this paper, we discuss our initial steps to develop a measure of students’ perceptions of instructor behaviors that promote inclusivity in their classrooms. This paper briefly discusses the initial development of items for our measure and focuses on our exploratory analysis of the latent factor structure and our process for defining these latent factors in the context of math instructors’ actions.
In this research study, we employed Schoenfeld’s (2010) theory of Resources, Orientations, and Goals (ROGs) to examine an instructor’s (and co-author’s) textbook in a second course in linear algebra. The course was proof-based and the instructor anticipated that many students would have difficulty. Having the knowledge of students’ difficulties from a first course in linear algebra they created tailored resources with many hints and pedagogical attributes for their students. The instructor’s goal was to encourage students to construct their own proofs to understand and learn the materials, hence promoting exploration, conjecturing, and proving mental mathematical actions.
Microaggressions (MAs) are intentional or unintentional messages that communicate hostile, derogatory, or negative messages towards a recipient (Sue et al., 2007). MAs that students receive in a math class can impact a students’ learning experience and can often lead to feelings of exclusion (Cawley et al., 2023). This paper expands on previous work, discussing two types of MAs—racial and gendered—while also discussing students’ overall sense of belonging in a math classroom. This study analyzes the reflections of 133 undergraduate math students who were asked to reflect on an article about mathematical MAs (Su, 2015). Findings show that a majority of students have felt like they do not belong in the math classroom, and that racial and gendered MAs contribute to this. This research supports the need to develop initiatives at departmental and institutional levels to encourage more inclusive spaces in math classrooms.
Studies show that Research-Based Instructional Strategies (RBIS) help students learn, however their adoption has been slow. The Teacher Centered Systematic Reform Model (TCRM) is a general model for organizing enablers and barriers to adoption of new teaching methods that includes departmental, personal and teacher thinking factors. We used the TCRM model as a framework to assess the amount of formal lecture reported by 634 mathematics instructors in their undergraduate courses. Regression analyses found that instructors who participated in Project NExT (a professional development workshop) during their early careers were less likely to use lecture than non-participants. Other significant predictors of lecture less included evaluation expectations emphasizing active teaching methods, involvement in equity and diversity efforts, and prior experience with RBIS. Factors with a positive correlational association with lecture included evaluation efforts by departments where lecture was expected. Results confirmed some prior models in different disciplines.
Research has shown how crucial quantities-based meanings are for calculus concepts. While past work has developed important quantitative approaches to integration, the major topic of u- substitution typically has not been fully detailed in these paradigms. This theoretical paper extends this past work to clearly define and elaborate u-substitution through a quantitative perspective. We use conceptual analysis based on quantities, starting with a concrete example of a solar panel producing energy. We abstract from this example to define u-substitution as a transformation from one quantitative relationship, via nested multivariation, to another quantitative relationship. We also detail a three-part structure within this transformation.
Within the line of work on students’ quantitative reasoning, researchers have alluded to the significance of time in constructing covariational relationships. I draw on this body of literature and return to Piaget’s perspective on time to provide a framework for the role of time in students’ (co)variational relationships. The framework also clarifies the nature of the multiplicative objects underlying students’ (co)variational relationships. In support of illustrating the framework and capturing its emergence from building second-order models of students’ mathematics, I also describe a task and how its design reflects the framework.
In this theoretical report, we outline several major traditions of genre theory, argue that proof qualifies as a genre as defined by these traditions, and propose how genre analyses could be used to further our field’s understanding of proof.
Students often express negative emotions in mathematics classes. Positive emotions feel good, but they are important beyond just feeling good. Positive emotions can benefit learning, attitudes, and perceptions of mathematics. In this paper, we discuss why positive emotions are beneficial along with a literature review of common themes which are associated with positive emotions among students. For example, increases in students’ perceptions of personal control and value in the classroom have been associated with increased positive emotions. Many practices already encouraged by mathematics education researchers (e.g. teaching conceptually, helping students find mathematics important) have the added benefit of potentially increasing positive emotions in students. Positive emotions have been shown to be associated with engagement and motivation in mathematics. By increasing positive emotions in the classroom, instructors can encourage the gradual growth of positive attitudes and beliefs about mathematics to students which are currently lacking among many students.
Play has been recognized as a component of mathematical practice; accordingly, this manuscript explores the potential role of play in undergraduate mathematics education. The manuscript introduces a novel framework for characterizing play, wherein a person is said to be playing in a scenario if they are (1) making non-routine, voluntary, and free choices/actions, (2) experiencing an appropriate level of challenge/uncertainty, and (3) experiencing amusement, satisfaction, or excitement. This framework is applied to provide a definition of playful mathematical experiences and consider how playful mathematical experiences relate to other concepts, like mathematical play and authentic mathematics.
Pairing and one-to-one correspondences are natural activities, which can be formalized using bijections. Despite bijections being central to mathematics, few studies have explicitly attended to how students construct, understand, and use them. Because bijections show up naturally in combinatorial arguments, I situate this work in combinatorics. In this theoretical paper, I propose three ways of understanding bijections in combinatorics—relabeling, representational, and relational bijections—and discuss possible affordances and constraints of each. I also provide combinatorial examples using binomial coefficients to exemplify each way of understanding.
Prior research on student understanding of inverse function has primarily focused on deficits in these understandings without explicitly addressing the mental actions and operations that researchers or instructors would like students to engage in. In this report, I present three ways of reasoning about inverse functions: a covariational approach, a mapping approach, and a set theoretic approach. I argue that all three entail productive ways of thinking about inverse functions, and that the coordination of the three can support students in reasoning about inverse functions in a variety of contexts.
In this paper, we motivate the need for a definition of mathematical autonomy that does not imply that autonomy is a trait of an individual. Rather, we demonstrate the usefulness of a lens on autonomy that captures the characteristics of actions taken in contexts that shape the space of sensible action. Shifting from trait to quality of action in context further allows us to analyze how actions of groups of individuals doing mathematical work can have qualities associated with mathematical autonomy. We ground our theoretical work in data from a longitudinal case study of an undergraduate student, Kaleb, whose orientation to mathematical sense-making changed sharply across multiple contexts of activity.
This theoretical report offers a conceptual analysis of expressing distances on graphs of functions in the Cartesian plane algebraically. We frame this mental activity as a connection between the algebraic and graphical registers, and describe three key connections it comprises: (1) differences express distances between two positions, (2) points are ordered pairs of distances from the axes in the Cartesian plane, and (3) equations give the relation between x and y for every ordered pair on a graph. We detail the conceptual components of each connection, which involve a component from each of the graphical and algebraic registers and an underlying interpretation that forges the connection between the two. This conceptual analysis makes explicit the complex cognitive steps involved in algebraically expressing distances on graphs of functions, with implications for researchers and practitioners using algebraic and graphical representations together.
Numerous studies have demonstrated that active learning can increase student learning and reduce achievement gaps; research has also shown that active learning in undergraduate mathematics is not consistently equitable. These findings highlight a gap in what we know about active learning and indicate the need for a deeper understanding of the relationship between equity, inclusion, and active learning. Drawing on research about inclusive and equitable mathematics learning environments across secondary and postsecondary contexts, in concert with what is known about active learning in undergraduate mathematics classrooms, I present a theoretical argument that active learning is a necessary but insufficient condition for mathematics learning communities to be inclusive and equitable. I close by suggesting potential strategies for ensuring active learning is implemented in ways that are inclusive and equitable.
In some classical semiotic perspectives, signs and sign systems can be viewed as either institutional or personal. This dichotomy can place personal representations as antecedent to institutional ones, sometimes with the explicit goal of transitioning away from the personal ones entirely. In this paper I draw from qualitative data to argue that personal representations are legitimate and sometimes necessary components of developing combinatorial facility, and they play an ongoing role in understanding that is not replaced by institutional representations. I frame this theoretical argument around a discussion of common institutional semiotic systems for combinatorics, which can insufficiently support a set-oriented perspective on counting. I discuss how to incorporate personal representations into analysis that uses a classical semiotic lens. Doing so includes attending to the motivations for creating (or refining) a sign system, the connections between those systems and others, and how the form and nature of the sign systems seem to impact or reflect student reasoning. I also discuss how some differences between combinatorics and other areas of mathematics might affect the personal/institutional dichotomy in semiotic analysis.
Isomorphism appears across many courses in mathematics, including discrete mathematics, linear algebra, and abstract algebra. However, examinations of student understandings of isomorphism have mostly focused on group isomorphism in abstract algebra. In this paper, we compare survey responses from 49 discrete math, 19 linear algebra, and 27 abstract algebra students who were prompted to explain the concept of isomorphism to a child. Results include a cross-course emphasis on types of sameness but variations in the types of language used to characterize isomorphism in each course. Implications include the need for care among instructors and researchers when referring the concept of “isomorphism” without a context as well as the need for further work in understandings of graph theory.
This paper presents emerging results from an investigation of undergraduate students’ beliefs about mathematics (and themselves as learners) within the context of a discrete mathematics course. Our data sources include findings from a survey study with 11 participants and a follow-up interview study with 5 participants. Results suggest that students perceived discrete mathematics as being significantly different from their previous mathematics coursework, offering additional options for creativity and a shift in focus from computation to proof and logic. The impact of this shift on students’ mathematical self-efficacy was mixed, with some students reporting unchanged self-efficacy but multiple students reporting decreased self-efficacy (either temporary or permanent). These results have implications for the teaching of discrete mathematics courses and shed additional light on our existing knowledge of students’ belief change during the transition from K-12 school to undergraduate mathematics courses.
Calculus has long been known as a “gateway course” to STEM fields in postsecondary education. To address this issue, researchers in the Math Department at Montclair State University designed a model of complementary instruction that features peer-facilitated workshops where Calculus I students work in groups on inquiry-oriented, groupworthy tasks. The purpose of this multiple-case study is to seek answers to the question, "How do undergraduate Calculus I students experience and navigate their learning of calculus in the parallel spaces of coursework and inquiry-oriented complementary instruction?" The findings of one case study are presented here and include characterizations of the different forms of agentive participation afforded to students in each of the two spaces, as well as their complementary nature relative to learning calculus with understanding. Implications for dismantling the persistent barriers imposed by calculus on access to postsecondary STEM fields are also discussed.
The current state of proof comprehension research has focused on the instructor or researcher perspective. Either through assessing student proof comprehension (e.g., Mejia-Ramos, 2017) or developing interventions to improve student proof comprehension (e.g., Hodds et al., 2014). In this report, we discuss findings from one student from an exploratory study. We consider the influence of prior experiences on what it means to them to understand a proof and the reasoning behind the proof comprehension strategies used to understand a given proof.
As part of an effort to examine students’ understanding of vector products, we present preliminary findings from student responses to survey questions asking whether the dot product has a direction in three different contexts and three different class levels. The contexts were two vectors with labels but no values, two vectors on a Cartesian grid with specified magnitudes and directions, and the flux of a uniform electric or magnetic field through a planar surface. The three courses were the second semester of an introductory calculus-based physics course, a sophomore-level mathematical methods in physics course, and a junior-level electricity and magnetism course. Depending on the context, between 25% and 30% of introductory students were successful at determining that the dot product has no direction. Performance increased with course level.
Student difficulties in introductory linear algebra courses are often attributed to the novelty of the concepts in the course and the disconnectedness of these concepts to students’ prior mathematical experiences. However, researchers have stated that a strong prerequisite understanding of the function concept is essential to students’ understanding of linear transformations in linear algebra. In calculus and related courses, covariational and multivariational reasoning has been determined to be necessary for students to appropriately reason about functions in two or more variables. However, research on multivariational reasoning in linear algebra, especially with respect to linear transformations, is scarce. To contribute to the literature, we designed a study exploring a hypothetical model of how students might come to reason multivariationally about linear transformations. In this preliminary report, I discuss a part of the study – interviews with seven mathematicians who have either taught linear algebra or used linear algebra in their research.
Teaching mathematics for future elementary teachers is fundamentally different from other forms of mathematics and thus requires different knowledge. As community colleges become increasingly involved in the process of training future teachers, it is essential to explore how instructors at these institutions develop as mathematics teacher educators. This paper reports on a preliminary exploration of how community college faculty grappled with teaching-oriented mathematical tasks involving fractions. Choices of mathematical representation, selection of answer before and after discussion, and overall themes are discussed, with a focus on development of mathematical content knowledge for teaching.
Mathematics Graduate Teaching Assistants (MGTAs) play a significant role in undergraduate mathematics education, as they are often in a teaching-related position for courses in the College Algebra through Calculus II sequence. Further, research shows that teaching practices that promote active learning, equity, and inclusion lead to improved student outcomes (e.g., Freeman et al., 2014; Laursen & Rasmussen, 2019; Mulnix, Vandegrift, and Chaudhury, 2016). However, MGTA PD that reflects these practices may be limited due to a variety of reasons. In this manuscript we discuss barriers and drivers to implementing PD that reflects active learning and equity-focused values. The goal is to understand ways that we can utilize institutional-specific drivers and navigate barriers to support the implementation of more PD programs like the one in this study.
We aimed to get a better understanding of participants’ (eight foundational math course [FMC] coordinators’) teaching approaches. In the first year of this grant project, we primarily gathered data (through surveys, self-reflections, and class observations) on these individuals as instructors. These data were compiled into narrative summaries for each participant and analyzed and compared. We discuss our findings from this analysis, using the instructional triangle as a framework, and particularly focusing on instructor-student interactions. This project aims to develop an understanding of what is needed to support instructional change in FMCs by evaluating how math-specific professional development (PD) cycles affect FMC coordinators’ teaching practices and perspectives. We seek audience feedback on potential next steps towards fostering effective instructor-student interactions and future PD cycles.
I was charged with developing and implementing a course redesign to improve low success rates in the lowest university-level developmental mathematics course (median ACT score = 17). As part of the initial implementation of the course redesign a diagnostic assessment was given to understand the nature of students’ conceptions of foundational concepts. Student responses on several items indicated that over 25% of students had a dominant part-whole conception of fractions; not seeing fractions in terms of measures. This dominant part-whole conception of fractions also influenced how students reasoned about items involving other concepts (e.g., addition of fractions and ratio comparisons). In this preliminary report the results that led to our conclusions, the instructional intervention we used to move students’ conceptions of fractions forward, and the impact of the intervention, based on the results of the post-assessment, on students’ conceptions of fractions will be shared.
Graduate education in mathematics is instrumental to the socialization of prospective mathematics faculty, however, our understanding of how graduate students develop their professional identities is still limited. This preliminary study examines and compares how two doctoral students at different stages of their graduate programs reflect on and understand their professional identities. I highlight qualitative similarities and differences in how each participant identified as a teacher, as a researcher, and as a mathematician. The structure of the program and progression through degree milestones reflected how strongly participants were anchored onto dimensions of their professional identities. Both students strongly identified as teachers and both described how their gender identity as women negatively impacted dimensions of their professional identities, especially their mathematics identities.
Feelings of inclusion in the classroom are one contributor to students’ decisions to pursue a particular degree pathway, and students from marginalized groups are more likely to experience feelings of exclusion in these spaces. These feelings may be exacerbated by particular classroom actions and interactions, including microaggressions based on race and/or gender as well as being ignored or talked over; these actions are often compounded by instructor and student biases. Being able to observe such events is key to instructors’ ability to disrupt those moments and develop more inclusive spaces, but many observation protocols are built from an external researcher perspective about practices that support equity. We report on a pilot study at an HSI, comparing what Hispanic and Latine students report makes them feel more or less included to what is typically captured by observation protocols.
Two mathematics professors at a large university were interviewed about how their emotions impact their research work. Joy and frustration were the emotions they reported experiencing most frequently. Both reported the benefits of collaboration and the importance of regulating both positive and negative emotions. Implications on collaborative learning in the classroom (e.g. groupwork, pair work) and implications for emotions in the classroom are discussed.
Previous research indicates that mathematics ability predicts student success in chemistry courses. Not surprisingly, chemistry students are required to take courses in mathematics as prerequisites and co-requisites, including calculus. In this work, we seek to identify the ways calculus skills and concepts appear in the undergraduate chemistry curriculum, focusing on the introductory general chemistry sequence. To this end, we combined relevant disciplinary frameworks and looked for alignment: the Calculus Content Framework (CCF) and the Anchoring Concepts Content Map (ACCM) for general chemistry. Preliminary results indicate explicit use of calculus is not common in introductory chemistry, although there are opportunities for instructors to use the general chemistry topic of chemical kinetics as a context for the application of differentiation and integration. Future work will involve mapping out the landscape of the remainder of the undergraduate chemistry curriculum and as an expected outcome of this work we seek to provide tangible resources to facilitate cross-disciplinary discussions between instructors in calculus and chemistry.
Learning Assistants (LAs) are undergraduate students who support active learning classrooms by acting as a second instructional figure in the classroom. Although research has shown a variety of benefits from the integration of LAs, little is known about the nature of the LA- instructor relationship and its impact on instruction. We present preliminary findings from a comparative case study of how graduate student instructors of record (GSIs) and undergraduate learning assistants (LAs) work together within an active learning classroom. Specifically, we focus on how the interactions between GSIs and LAs relate to the professional obligations felt by GSIs and LAs, both of whom are positioned as mathematics instructors, albeit in different ways. This preliminary report highlights initial findings and suggestions for further investigation.
We explored the intersections between two sets of pre-service teachers' beliefs, revealing potential relationships between beliefs about mathematics and beliefs about EB mathematics education. PSTs whose beliefs about mathematics were consistent with reform-oriented mathematics were identified as social constructivists, and they were more likely to express equitable beliefs about EB mathematics education. However, we find a more mixed pattern regarding beliefs about EB mathematics education among PSTs identified as moderately-traditionalists. Specifically, we find that the association between PSTs’ demographic backgrounds and engagement with EB issues and their endorsement of more moderate beliefs may differ across different groups of PSTs. This research contributes to understanding the complex interplay of teacher beliefs in mathematics and EB education, offering insights that can inform more effective teacher-preparation programs to enhance PSTs' ability to teach EB students effectively.
In what ways do students bear mathematical authority in advanced mathematics classrooms? What authority relations arise within inquiry-oriented classrooms? Answering these questions may give insight into the inequities that exist within undergraduate mathematics education. We share findings from our initial investigation of 7 lessons from one inquiry-oriented abstract algebra (IOAA) classroom to determine who gets to have authority to do certain activities. Taking an action-based approach to parsing authority relations, we attended to who bears authority for the activities of authorship, communication of ideas, and assessment of ideas. Our preliminary findings propose five categories of activities for assessment as well as several observations about the nature of authority relations within an IOAA classroom.
There has been a recent trend in the application of fine-grained constructivist frameworks such as the resources framework; however, the resources framework has been operationalized in a variety of ways, particularly regarding analytic choices. In this work I seek to further discussions surrounding theory use and the ways frameworks may inform (1) research questions and (2) data analysis. Using an analytic autoethnographic approach, I characterize the variation in the application of the resources framework in my own work (n=10). This allows me to provide specific examples without evaluating the quality of colleagues’ research, as well as contextualize the research within my experiences and within the broader norms of the chemistry education research community. To this end, I discuss alignment between the research questions and the resources framework and demonstrate variation in how “cognitive units” were operationalized. As an implication, suggestions are made regarding future work using the resources framework.
Students often have to interpret conditional statements in mathematics classes without prior training in logic. In a previous study, we investigated how students interpreted variants of the conditional statement: If two random variables are independent, their covariance is 0. The purpose of the current study is to explore surprising results and apparent inconsistencies in student interpretations using clinical interviews after a lecture introducing the conditional statement. In this paper we contrast the experiences of two students, Byron and Sophie. Preliminary results suggest that Byron relies both on his logic training and his domain knowledge to solve problems about independence and covariance. Sophie’s incorrect domain knowledge led to her reversing the implication. Once this was resolved, she was not confident in assigning truth values to statement variants. We discuss teaching implications that arise from issues of both mathematical efficiency and accuracy in presenting this theorem.
While there is evidence that college students benefit from classroom activities that engage them, a shift in instruction from the lecture style of teaching to methods where students are actively involved in their learning is difficult to generate. This study focuses on providing structure and support for college algebra instructors to implement evidence-based instructional practices. We interviewed participants each semester they were involved with this project to gain insight into their perspectives on their teaching practices. We also video-recorded three lessons each semester when participants were implementing co-constructed in-class materials. Initial results showed how collaboration empowered instructors to implement evidence-based instructional practices in their class. Additionally, even though instructors collaborated to co-create in-class materials, results showed variation in their implementation styles as they moved towards more student-centered practices.
Proof and proving are critical to the work of mathematicians, and in turn research on the teaching and learning of proof has historically been one of the larger strands of research presented at annual RUME conferences. In this project we systematically reviewed the RUME conference papers on mathematical proof from 2018-2023 to provide a picture of (a) the most prevalent topics of proof research in RUME, and (b) the data sources drawn upon in proof research methodologies in RUME. This review revealed the most frequent primary topics of RUME proof research to be proof instruction, assessment and feedback, and proof conceptions. We also noticed an overwhelming proportion of RUME proof research papers rely on qualitative data sources. Our continued work on this project will allow us to examine the impact research RUME has on the larger body of knowledge of undergraduate proof research.
Team-Based Inquiry Learning (TBIL) is a novel active learning pedagogy designed to facilitate the use of inquiry-based learning in lower division courses. This preliminary report examines supports provided by the TBIL project to instructors, as well as the fidelity of implementation of TBIL by participants of the project. Initial findings suggest that classroom-ready materials and ongoing support, both synchronous and asynchronous, were most helpful to faculty in their TBIL implementations.
Mathematical proof writing is known to follow various norms and conventions set by the mathematical community. Yet, recent scholars have questioned if these norms and conventions hinder the progression of the field to be inclusive in proof courses. In this report, we discuss preliminary findings of a linguistic analysis on the degree of gendered language used in eight undergraduate textbook proofs ranging in content. We report a mixture of gendered language use in mathematical proof writing.
Linear algebra is an important topic in mathematics and many other disciplines. In this paper, we consider a set of digital interactive figures (I-figs) using Mathematica created for linear algebra students in an introductory course. The figures were designed to facilitate students’ ability to visualize and work with eigenvalues and eigenvectors while minimizing computation for the benefit of conceptual focus while looking at an unlimited number of examples. The worksheets provide a foundation for motivating students to participate in a system of observation, conjecture, proof, and theorem. Based on our preliminary analysis of students’ reflections on the I-figs, we found that students were confident in making and believing example- based conjectures.
Discussion of a study that examines data from a single large, public university to determine how and when precalculus students elect to use learning center resources and their experiences using these resources. Data collected and to be analyzed includes survey and interview data from the precalculus students, interview data from instructors, class and learning center artifacts, and interview data from learning center leaders.
Student described experiences in creativity can provide insight into instruction to foster creativity. In this report, I describe the emergent themes among student narratives describing when and why they felt creative during a collaborative proving activity. Continued work to refine these themes and their prevalence in student reflections can provide implications for how to best structure collaborative work to foster creativity in proof-based courses.
A student’s course grade is typically an overall representation of multiple graded categories chosen by faculty. This study uses interviews with math and physics faculty to determine their rationale for selecting grading and assessment systems. We hope by identifying common faculty motivations, we can provide a baseline for future discussions on the implementation of alternative grading practices. Semi-structured interviews were conducted to present math and physics faculty with questions regarding enrollment size, length of implementation, and feedback processes. Preliminary analysis of faculty interviews demonstrates significant similarities in rationale across departments. All faculty considered enrollment size and course content when selecting systems. Findings indicated two central student learning goals and the emergence of four dominant motivations.
We present work in progress that addresses the question: How do researchers in mathematics education define and operationalize the concept of equity in their work? Using a wide search of articles published in mathematics education research journals over the 2000-2022 period, we identified an initial sample of 47 empirical and theoretical articles, which was then reduced to 28 articles that operationalized or defined equity. In 17 of the 28 articles that included a definition, we found that equity definitions encompassed mainly notions of fairness, access, participation, and demanding mathematics, and we observed several references to bridging, identity, agency, and power. We summarize the scholars’ rationales, purposes, and motivations for addressing equity across the 28 articles and suggest that a more critical stance is needed in the operationalization of a definition of equity.
DFW rates in first-year mathematics courses are a major concern for undergraduate institutions. In this study, we explored statistical trends in grades for students who repeat mathematics courses or transition to subsequent courses. We analyzed trends among final grades in courses from Intermediate Algebra through Calculus II over the past five years at a major midwestern metropolitan university. Our results show that students are statistically more likely to pass a class on their first attempt, and that DFW rates among students who repeat a course hover around 50% for subsequent repetitions of the same course. We also found that final grades are a strong predictor of grades in subsequent courses, with students who earn an A in the previous course being 3 and 4 times more likely to pass the subsequent course than students who earn a B or C, respectively. We conclude by discussing the implications of our results.
A blended course model uses both classroom learning and online learning to create flexibility in the weekly schedule for students. In this paper, we describe a blended course model and the rationale for implementing it in Calculus I. We report on student responses to the model and the behaviors it encouraged. The design of the blended course model created the opportunity for office-hours-like interactions in scheduled class time to help reveal the hidden curriculum of office hours. Two goals the model strived to achieve for students were (1) normalizing help-seeking behaviors and (2) normalizing working with peers. Data collected through questionnaires and focus group interviews suggest that the blended course model achieved these goals and more.
The Charles A. Dana Center worked across all public colleges and universities in Arkansas through a project called Strong Start to Finish (SStF) Arkansas to support the implementation of corequisite support for underprepared students in mathematics and multiple mathematics pathways for all. Consistent with Reinholz, White, and Andrews’ (2021) call to more effectively ground change efforts in change theories, the Dana Center designed its innovation using a change theory-based theory of change. This preliminary report describes this project’s outcomes and how the projects’ results further the field’s understanding of Rogers’ Diffusion of Innovations theory, one of the change theories on which the project’s theory of change was based.
Studying student experiences in undergraduate mathematics education serves as a fundamental aspect of understanding how mathematics departments can best serve the most at-risk populations of students who are often harmed by racial and gender inequities present in student success rates in undergraduate mathematics courses. In this study, we focus on support courses for Precalculus and Calculus I, where students engage in frequent Supplemental Instruction (SI) sessions and an online course run by the university Math Equity Coordinator. In analysis of a mixed-methods approach consisting of interviews and a survey, we examine the impact of these supports on students' sense of belonging and confidence in mathematics. Preliminary results from survey analysis show that students in the support course report higher levels of belonging and confidence in mathematics. Furthermore, interviews highlight the importance of instructor mentorship in enhancing students' confidence and sense of belonging.
I describe findings from a hermeneutic phenomenological study of white male engagement in DEI faculty service. I frame the phenomenon of interest using the DEI Institutional Activism Framework, and questions of structure and agency through Strong Structuration Theory. Two major findings are reported here. First, that DEI service is disincentivized by contradictory institutional structures that devalue engagement. Second, that individual faculty agentically engage in DEI service because of relational connections to students, but also to women they know personally (daughters, wives, friends, colleagues). These preliminary findings are based on semi-structured interviews with four STEM faculty.
We report on preliminary findings from a study of the ways in which linear algebra students used function composition to describe the result of applying two linear transformations, defined using symbolic vector notation, to a graphical region. Our current results from even a small set of data suggest that the students engage in function composition in a variety of ways. These results suggest areas for future exploration regarding how students engage in these forms of composition with linear transformations, and how their conceptualizations of individual transformations inform their understandings of the composition.
The corequisite model of academic support has been touted as an efficient way of supporting student learning while decreasing the time to degree completion for students needing academic support. With the learning losses that have occurred globally as a result of the global coronavirus pandemic, the Department of Mathematics at California State University, Fullerton piloted a corequisite course to support one section of Calculus I, a course historically riddled with high DFW rates. In this paper we share preliminary results of our department’s implementation of the corequisite model. While corequisite students from the first iteration of the course earned higher GPAs than their non-corequisite peers, more data (including longitudinal data) is needed to determine the lasting impact from the Calculus I corequisite experience.
Modern statistics education requires that we support students in building powerful and productive ways of computational thinking. In this paper, we seek to understand the ways of computational thinking that undergraduate students employ as they engage with data. To address this question, we administered task-based interviews with three participants using the R programming language. Problem-solving approaches focusing on formatting data and efficient coding emerged as early aspects of student thinking. We are still reviewing interview transcripts and intend to compare our findings with existing frameworks from the literature, working towards a framework highlighting beneficial ways of thinking. This work is part of a larger study with additional tasks and additional individuals with varying levels of experience and expertise.
The secondary-tertiary transition (STT) in mathematics presents university students with multiple cognitive-epistemological, didactical, sociocultural, and affective challenges. This qualitative study explores the perceptions of two pre-service mathematics teachers regarding the relevance of abstract mathematics to their future teaching practices. Using three interviews and eleven reflections, the investigation yields in-depth insights. In addition to highlighting the affective aspects of the STT in mathematics, the study addressed certain difficulties pre-service teachers encountered while dealing with advanced mathematics courses.
Function is a unifying, cross-curricular theme that plays a central role in nearly every subdiscipline of mathematics. Yet, this cultural meta-narrative that is widely accepted by mathematicians is not one students readily adopt. In this report, I share a preliminary analysis of the stories told about three different types of multivariable functions in a commonly used calculus textbook. In doing so, I juxtapose these stories and consider how this textbook—a cultural artifact of the discipline of mathematics—might support or hinder the transmittal of desirable cultural meta-narratives about the role of function. Ultimately, I find that real-valued functions are consistently positioned as functions, while the function properties of both types of vector-valued functions are de-emphasized. As only one of the three function types are positioned as functions, this suggests that the stories in the textbook could hinder students’ adoption of the meta-narrative of function as a unifying mathematical theme
As part of a larger project to investigate student use of mathematics in upper-division physics courses, we have examined how students use series expansions as a means of approximating and simplifying complicated expressions in theory-oriented physics courses. Student responses were collected on pre- and post-instruction written tasks in which students were prompted to use a series to approximate an expression arising from a problem in electricity and magnetism. Despite prior experience with series in calculus and physics, students struggled to determine appropriate quantities in which to expand and did not attend to units or the convergence of their series. While student success rates improved after targeted instruction, many students expanded in quantities that were neither dimensionless nor small, and thus unproductive for modeling.
In this study, we looked for how textbooks used in geometry content courses for elementary school teacher candidates might support their pedagogical content knowledge. The conceptual framework of this study is Ball et al.’s (2008) practice-based theoretical model that refines pedagogical content knowledge which blends content knowledge, pedagogy and curriculum that highlights teacher’s knowledge of standards, grade levels when particular topics are taught, and more. We collected the common used textbooks reported by U.S. mathematics courses instructors for elementary teachers (Max & Newton, 2017; Authors, 2023). Drawing from relational content analysis methodology (Elo et al., 2014), we first analyze the content of five geometry textbooks and identify the instances of pedagogical content knowledge elements while considering the conceptual framework and examine the differences and similarities across the textbooks. Analysis revealed that of 780 total codes, the highest percentages of codes focused on children’s ideas which provides many examples of how children reason through geometric concepts. The second-highest code percentages focused on the teacher’s role and classroom connections, such as discussing teachers’ responses to children’s ideas to advance the latter’s geometric reasoning. The third-highest percentage of codes focused on the use of learning standards, which was often listed next to each relevant book session to show how standards could be linked with instructional activities in progressing through grades K–5. The fourth-highest percentages were related to the code for children’s written work which would allow prospective teachers to see concrete examples of children’s reasoning through geometric concepts. The code with the least frequency related to the use of assessment questions, providing sample questions from actual standardized tests for K–5 students to familiarize prospective teachers with the content for different grade levels.
Empirical studies on mathematical convictions often report on proof comprehension or evaluation. An underreported method in this literature is script-writing tasks. This method is often used with teachers as lesson-play while others have used it for proof comprehension. The purpose of this study was to explore how this method may elicit students' mathematical convictions. The guiding research question was, how do students' responses to a script-writing describe their mathematical convictions? Two undergraduate students' responses showed that their convictions remained the same throughout the study. Their scripts showed that they may help another student by helping another understand or clarify the proof method or convince them of the statement's truth. Future research may explore more complex statements and arguments using this method of research.
Carlson et al. (2010) developed the 25-item multiple choice Precalculus Concept Assessment (PCA) to investigate reasoning abilities and meanings researchers (e.g., Carlson et al., 2002; Dubinsky & Harel, 1992; Oehrtman et al., 2008; Thompson & Silverman, 2007) have established as critical for pre-calculus and calculus learning. Since its initial development and validation, the PCA has been administered to thousands of secondary and post-secondary students, providing key insights into their reasoning abilities, as well as their potential success (as measured by grades) in future calculus courses. Given the insights the PCA provides relative to pre-calculus and calculus students, we grew interested in the extent the PCA can provide useful insights with other populations. Across several semesters, we administered the PCA to 174 undergraduate students upon their entry to a secondary mathematics teacher preparation program. In this poster, we present on the results of that administration. Specifically, we report on an analysis of the aggregate scores of the population, as well as their performance on covariational reasoning items. With respect to the covariational reasoning item performance, we draw on our expertise and research (Moore, 2021; Moore et al., 2022; Moore et al., 2019) to develop hypotheses regarding discrepancies in performance.
This study reports on first-semester calculus students’ reasoning about a univariate optimization problem that involves finding the production level at which the cost per yard is minimized when given the graph of a function that represents the relationship between the cost per yard and the number of yards produced by a factory. Analysis of verbal responses and work written by four students when solving the problem revealed that determining the production level at which the cost per yard is minimized was straightforward for all the students. However, explaining how this production level is related to the first derivative of the given function was problematic for most of the students.
This exploratory study investigated the relationships between professors’ enactments of a research-based precalculus curriculum (Pathways) and changes observed in students’ attitudes towards mathematics and perseverance in problem-solving. While much research focuses on improving student achievement in undergraduate STEM courses, it is also important to support students in developing the positive dispositions and practices needed to sustain them through years of mathematics-based STEM coursework. Study participants included three precalculus professors and 33 of their students. Data collection and analysis involved classroom observations focused on each professor’s enactment of the curriculum, surveys of student attitudes towards mathematics, and measures of student perseverance during problem-solving sessions. Our findings suggest that although the Pathways precalculus curriculum may support the development of positive attitudes toward mathematics and improved perseverance in problem-solving, this potential is influenced by professors’ pedagogical choices.
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A mathematics peer tutor verbally expressed belief in a growth mindset, but also expressed beliefs that students who don't succeed are simply not putting in the effort. This poster reflects on the juxtaposition of these beliefs and their implications for helping students succeed in mathematics. Where does motivation come into play? Can students grow in motivation and academic skills just as we believe they can grow mathematically?
Every year, millions of students begin post-secondary education in the hopes of furthering their education, graduating with a degree, and landing a good job in the future. However, research shows that being college eligible is not the same as being college ready, and first-time students struggle with the rigors of higher education. This can affect student success, both academically and psychologically. To help students acclimate to college life and academics, universities and colleges have created student life skills (SLS) courses which focus on nonacademic skills and learning strategies. While research has shown that SLS courses provide a wide array of benefits to students, these courses often attend to general concepts and techniques without considering their application to different subject areas, making the process abstract for students. One discipline that could highly benefit from incorporating SLS content into the curriculum is mathematics, as certain mathematics courses in higher education are considered gateway courses to advanced coursework for many majors. In this poster session I will discuss how I created a special Learning to Learn Business Calculus workshop focused on nonacademic and mathematics strategy skills, the different components and activities of the workshop, and the ongoing research study which is measuring its effectiveness and usefulness (from the student perspective) as an educational intervention.
In this poster we analyzed the mathematical autobiographies or mathematics stories from our University's 'Summer Launch' program for incoming freshman to take their entry-level (e.g., College Algebra) mathematics course with corequisite support. By combining frameworks from Watson (2019) and Drake et al. (2001) we found that students' stories typically fall under four categories: familial, antagonist, friendly, and roller-coaster (frenemy). We outline these categories, and hope to discuss how instructors can leverage students' stories to foster learning and improve mathematical attitudes.
In recent years, research on professional mathematicians' mathematical practices, including how they understand mathematical concepts, has been become more common (e.g., Oehrtman et al., 2019; Shepherd & van de Sande, 2014; Wilkerson-Jerde & Wilensky, 2011). Closely related to understanding is the notion of abstraction. Likely the most prominent theory of abstraction is Piaget’s reflective abstraction and his broader genetic epistemology (Piaget, 1970). Using Piaget's broader theory, this study introduces three different theoretical levels of understanding: pseudo-object-level, process-level, and object-level. With these levels of understanding, this study addresses the following two research questions: (1) In what ways do professional mathematicians operate with highly-abstract, advanced mathematical concepts at different levels of understanding? (2) What factors can influence a professional mathematician’s level of understanding for a given mathematical concept? In this study, I interviewed six professional mathematicians with three distinct semi-structure interviews. Data analysis revealed that mathematicians operate differently depending on their level of understanding for a mathematical concept. Moreover, various factors influence mathematicians' level of understanding for a given mathematical concept, including how the concept is being utilized and certain sociocultural factors.
Series convergence is a key part of the calculus curriculum; however, there is limited research on ways to support students conjecturing about series convergence. We conducted a teaching experiment using Realistic Mathematics Education in which we supported two undergraduate calculus students in reinventing statements about series convergence. We present the students’ reinvention of the comparison test and how it was related to their informal ideas about the sequences n^-1 and n^-2. Our study is an existence proof showing that students can be supported to conjecture tests for series convergence.
This poster focuses on the results from the second stage of a larger qualitative study designed to identify the mathematical connections preservice teachers make between abstract algebra and secondary school mathematics and explore what tasks and activities help them form those connections. Mathematical tasks involving secondary mathematics content were implemented as part of the curriculum in an abstract algebra course with undergraduate and graduate students. Six undergraduate students with interests in mathematics teaching at the secondary level and a graduate student with secondary teaching experience were the main participants of the study. Through the instructional tasks and additional interviews, all seven participants were able to see mathematical connections between abstract algebra and secondary school mathematics. Results suggest that abstract algebra faculty can support preservice teachers in abstract algebra by simply providing opportunities throughout the course for them to explicitly explore connections.
Epistemological Obstacles (EOs) are defined as challenges that students face when exposed to concepts that contradict their pre-existing mathematical intuition (Norton & Arnold, 2023). The objective of our poster is to address EOs that emerge during the lesson on proving by cases, rather than circumventing them. This approach, as pursued by Norton and Arnold (2023), elicited EOs such as the Principal of Universal Generalization (PUG) and understanding the difference between the role versus the value of a quantified variable (Qrv). PUG entails proving “for all” statements with an arbitrary value. It was mainly encountered during mathematical induction by Norton and Arnold (2022), but its relation to proofs by cases was not studied. When proving statements with PUG, it may not occur to students that they need specific (x=0), or arbitrary (x is even) cases. Qrv teaches students to prioritize maintaining generality; however, loss of generality is allowed with proof by cases as it is preserved in the aggregate. The obstacle comes from students struggling to understand that generality is maintained differently depending on the situation. The two EOs are closely related; Qrv can be thought of as a direct product of PUG. Our poster aims to elaborate on our proof by cases lesson and get feedback on how to revise the lesson in preparation for us submitting a journal manuscript on EOs in proof by cases
In this poster presentation, we present pilot data that emerged from working with secondary math teachers enrolled in a fully online, Introductory Real Analysis course in the Southeastern United States. This pilot data will inform our second attempt at implementing the recommendations of the Project ULTRA (Wasserman et al., 2022) research group in an online context with in-service teachers in a future academic semester.
Despite the importance of proof that plays in teaching and learning mathematics, pre-service secondary mathematics teachers’ difficulties with proof are well-documented. One of the primary challenges pre-service secondary mathematics teachers (PSMTs) face in learning proof is that they might not appreciate various roles of proof in mathematics. Given that PSMTs play a critical role in shaping their future students’ experiences with proof and proving, we seek to better understand how PSMTs’ interpretations of the verification, explanation, discovery, systemization, and communication roles of proof align or misalign with that of the discipline (as represented in De Villiers’ (1990) seminal work).
It is important for preservice teachers (PSTs) to have a deep and connected conceptual understanding of mathematical concepts they will teach in their future classrooms (Ball, 1990; Da Ponte & Chapman, 2015). Therefore, it is important to investigate PSTs’ conceptions, connections, notions, and understandings of specific mathematics concepts so that results can be used to inform effective instruction in content courses for future teachers.
There is limited research in undergraduate mathematics education on the relationships between embodiment and learner’s affect. In this poster presentation, we discuss how a summative assessment can help linear algebra students develop their own language for abstract mathematical concepts (that were introduced via various embodied activities throughout the semester) and bring their identities into the undergraduate mathematics classroom. Specifically, we report on a final exam an instructor used in her linear algebra class. The exam consisted of a written and oral component. In the written component, the students reflected on how the embodied activities supported their learning of the concepts and created metaphors for these concepts (e.g., null and column space, linear independence, determinant, eigenvector, dot and cross product, etc.). In the oral component, the students shared their metaphors in a focus-group setting. Preliminary findings suggest that the classroom embodied activities informed the students’ metaphors and thus shaped the students’ reasoning about linear algebra concepts. The students also exhibited behavioral and affective domains through embodiment such as using their bodies to portray their imagined objects and to convey their joy and frustration with creating metaphors. We conclude our presentation with some practical implications of our work.
Proof is vital to the work of mathematicians. We investigated how students in an introduction to proof course first come to understand the aspects of proof described by de Villiers (1990) when validating mathematical proof. Students in an introduction to proof course were put in small groups and asked to review sample arguments at various points throughout the semester. We use the definition of proof provided by Stylianides (2007) and focus on how students attended to the set of accepted statements (S), modes of argumentation (A), and modes of argument representation (R). Findings of this poster will include student quotes highlighting each of the aspects of the definition given by Stylianides as well as which aspects were prioritized at each point in the semester.
The Proofs Project is an NSF-funded research project investigating persistent challenges students experience in Introduction to Proofs courses. These challenges are framed in terms of epistemological obstacles that students and instructors experience during classroom interaction, even within best (research-based) practice. The project has generated video modules illustrating such obstacles and tasks that instructors can use to begin eliciting and addressing them. Now, in the second year of the project, instructors at three different universities have begun using project materials in their Introduction to Proofs courses. These three instructors participated in a three-day Proofs Project workshop at Virginia Tech, during the Summer of 2023. Here, we report on their experiences implementing project materials to intentionally elicit and address obstacles they have experienced in prior instruction, or anticipate experiencing in future instruction. Data come from surveys conducted at the end of a summer workshop; at the start of the following Fall semester, as they taught their next (or first) section of Introduction to Proofs; and at the end of that semester. Findings from qualitative analysis of the surveys indicate that all three instructors were able to use project materials to elicit—and begin to address—epistemological obstacles identified in prior research. In addition to discussion of specific findings, the first three authors will report on their individual experiences in implementing project materials to elicit and address persistent challenges students and instructors experience in Introduction to Proofs courses.
Machine Learning classification models can be used to algorithmically extend the qualitative coding mathematics education researchers commonly complete. This methodological poster provides a systematic way to conceptualize the qualitative coding process to leverage machine learning. We also provide an example of how feature analysis, a critical human component to developing a machine learning algorithm to classify data, can be completed with qualitative mathematics data. The poster will also include a brief discussion on the benefits and limitations of integrating machine learning into mathematics education research.
Calculus students often struggle to understand the derivative conceptually, even when they can differentiate fluently. Operating from the premise that underdeveloped covariational reasoning skills may be the missing link, this study explores how two widely-used calculus textbooks’ introductory derivative materials vary in terms of the opportunities they present for students to reason about the derivative covariationally. Using a novel framework introduced by Tasova et al. (2018) that merges Thompson & Carlson's (2017) covariational reasoning framework with Moore and Thompson’s (2015) graphical shaping thinking framework, this study describes the nature and frequency of the opportunities students have for understanding the derivative as dynamic or static. This poster describes the framework, presents the results of this analysis, and offers suggestions for both research and practice aligned with promoting a dynamic understanding of derivatives, and discusses the implications of these results for equity and inclusion in STEM.
We examine how embodiment-focused teacher noticing may contribute to rehumanizing mathematics assessment. We outline our approach as mathematics teacher educators (MTEs) wherein we investigated how prospective mathematics teachers (PTs) notice students' embodied actions with a focus on how (a) embodiment impacts assessment, and (b) assessment through embodied noticings can contribute to a rehumanizing of mathematics. The study emphasizes the need to overcome deficit framing and narrow interpretations in teacher noticing, encouraging teachers to consider students' embodied evidence and value mathematical actions beyond traditional algorithmic perspectives.
This theoretical poster engages with and presents dominant undergraduate mathematics topics as lively forces with rich potentialities for connection beyond disciplinary mathematics. More specifically, this poster offers both (a) a theoretical contribution to engage with undergraduate mathematics as situated, open, and lively (de Freitas & Sinclair, 2013; Mikulan & Sinclair, 2023); and (b) a potential bridge to resurface and generate connections between undergraduate mathematics and philosophical ideas that are often left out of mathematics classrooms. For example, we can use the concept of isomorphism to invite wonderings such as: where does the persistent occupation for ‘sameness of forms/structures’ in dominant mathematics come from? We invite our field to, as Mikulan and Sinclair (2023) write, care for an abstraction: “To care for an abstraction is to make sure you haven’t extended it too far, forgotten the contingencies on which it depends, applied it flippantly or use it to foreclose thinking” (p. 70).
This poster reports on a state-wide effort in California to reform post-secondary discrete mathematics curriculum and instruction. The focal research examines student and faculty experience of seven team-worthy lessons used as substitutes for lecture and of associated scaffolds for faculty on equitable instruction using the lessons. Early results indicate increases in students’ sense of access to, and engagement in, the intellectual work of discrete mathematics, growth in instructor knowledge of equity-supportive practices and in perceptions of themselves as facilitators of learning, and a positive reshaping of instructors’ views of students’ capabilities.
In this poster we illustrate how stewardship, a particular kind of leadership, in the complex system of mathematics instructional development requires decentering and interconnecting. This theory development for professional growth of faculty agents for change expands on earlier work describing how instructional practices used by providers of teaching-focused professional development in seminars about teaching (for graduate students) could be beneficial both for learning high-powered approaches to teaching of undergraduate mathematics and for building a foundation for future change-agent work. Here we move one level up and present an analogous argument about practices for stewards who are teaching about teaching about teaching. The poster illustrates the multilevel system with an expanded model that incorporates learning objectives for provider professional learning and the instructional practices of such professional learning in ways that showcase (and teach about) decentering and interconnecting.
Experiences in first year mathematics classes predict persistence in a STEM major (Seymour & Weston, 2019). Retention becomes increasingly difficult when students need to develop foundational skills in prerequisite courses such as College Algebra. The development of study skills, including metacognitive skills, are often used to improve low success rates. As part of a study on the impact of metacognitive instruction for College Algebra students, we found that when reflecting on the reason for their errors, students often attributed exam errors to “simple mistakes.” Researchers identified many of these mistakes as “not simple.” Classifications of “simple” or “not simple” mistakes by undergraduate peer tutors, who provide support in campus learning centers, did not consistently align with either the views of students or researchers. We discuss student, tutor, and researcher views of mistakes and how they compare with each other.
We discuss ways that one woman used learning resources during her university calculus course. We attend to connections between her resource usage and her mathematical identity by highlighting specific quotes. We argue that these connections provide evidence that her learning resource usage was more complex than it may seem on the surface.
In this poster, we share preservice elementary teachers' (PSTs') initial noticings during exploratory data analysis of materials focused on developing content knowledge of statistics as well as normalizing conversations of race in the mathematics classroom. The content covered in the module is typically part of a college-level introduction to statistics courses. We summarize PSTs' noticings and wonderings of an interactive data dashboard presenting racial disparities in school discipline. Our data reveals that without nudging, none of the PSTs discussed racial disparities in school discipline. Our findings suggest that PSTs may be uncomfortable discussing racial disparities in school discipline, emphasizing the importance of addressing these issues in mathematics teacher education and the need for mathematics teacher educators to provide guidance and support to help them connect mathematics with social justice issues. Future research will explore the impact of different nudging techniques in this context.
The retention of students in STEM fields has garnered significant attention in recent research, with a focus on understanding and improving student beliefs in mathematics to increase academic success. However, there remains a gap in the literature regarding the beliefs of first-generation college students, who often face unique challenges. This study investigates the mathematical beliefs of first-generation calculus students using the Indiana Mathematics Beliefs Scale (IBMS) and explores their potential implications for STEM retention.
Undergraduate students enrolled in introduction-to-proof courses often experience persistent challenges (i.e., epistemological obstacles) in learning core proof-related concepts, including logical implication and quantification. Our ongoing NSF-sponsored project examines students' experiences in such courses from multiple vantage points: whole-class discussions, small-group discussions, and clinical interviews with individual students. In this poster presentation, we discuss findings from our analyses of the discussions of one small group of students (the "Triangle Group"). Three themes emerged from our analysis. First, we found that students often place additional constraints on the truth set of a mathematical statement. Second, we found that students may not distinguish between the negation of a statement and a counterexample to that statement. Third, we found that small-group discussions can support students to address epistemological obstacles. Examples of each theme are provided in our presentation.
The proposed poster describes information collected from a case study of a DEI initiative in the math department at Kappa University (KU). A Networked Improvement Community (NIC) of faculty members and administrators began meeting monthly to discuss DEI in introductory math at KU. NIC leaders solicited applications from undergraduate and graduate students. Application responses were used as data to answer the question: What motivates students at KU to get involved in DEI work in the mathematics department? How do students bring their identities into these motivations?
This poster presentation intends to present, discuss and show ways to prepare graduate students better to teach in two-year and four-year institutions post-Covid (2020). Trends show that there are and will continue to be gaps in conceptual understanding of the mathematics concepts of our enrolling students in post-secondary institutions. This poster will present varying teaching models and professional development for mathematics graduate teaching assistants intending to teach at two-year and four-year institutions. Topics will also include adapting teaching in various mediums (face-to-face, Hybrid, synchronous, and asynchronous online) to address student needs, institutional needs, and expectations. This session aims to provide other graduate faculty mentors/coordinators with additional ideas, support, and aid in supporting the growth of their graduate students to become well-prepared mathematics educators.
This qualitative study aims to contribute to the understanding of developmental mathematics (DM) instructors by exploring the experiences, perspectives, and training of instructional stakeholders of a mid-level, coordinated, algebra developmental mathematics course (DMC). In semi-structured interviews, participants are asked to describe the purpose, goals, and outcomes of the DMC. Participants include instructors of various levels (i.e. graduate students, adjuncts, and non-tenure track instructors), the course coordinator, and departmental figures involved with instruction. Conducting this investigation with coordinated DMC stakeholders allows for the consideration of the impact of course coordination on in-class instruction and how/if stakeholders’ perspectives differ depending on their role in the department. An aim of this inquiry is to inform future work exploring the experience of DMC students, their learning in DMCs, and their broader attitudes towards math.
There is a strong need to support mathematics graduate teaching assistants in learning how to teach using active learning in ways that are inclusive and equitable. Given research on how active learning can exacerbate classroom inequities, it is crucial that we assist graduate students beyond solely active learning techniques. We report on a multi-year professional development for graduate students, as instituted at one site. We examine: How do MGTAs’ responses about teaching with active learning shift through a year of sustained professional development? We track three MGTAs’ responses over time, for their individual trajectories. Analysis reveals increased sophistication in articulating their thoughts regarding teaching, centering of students, and the cultivation of intellectual need for learning more about and tools regarding issues of diversity, equity, and inclusion in teaching.
Our research investigates the relationship between group dynamics and mathematical learning, specifically within the context of undergraduate math students working in small groups to complete an “Interpreting Graphs for Calculus” activity. We analyzed 10 videos of 2-3 students working together during one 50-minute class period and their written work. We developed a preliminary analytical framework to consider both knowledge-based and social dynamic features of the interactions we observed. Two prominent themes emerged across groups in the interactions: the role of pattern recognition and students’ willingness to correct their peers.
Some challenges of teaching General Education Mathematics Courses (GEMC) motivated us to investigate how doing collaborative projects affects undergraduate students’ self-efficacy in mathematics. The problems we see in the GEMC include students’ low self-efficacy in mathematics, low engagement, and negative attitudes toward mathematics. It is necessary to understand better how collaborative learning approaches work related to students’ efficacy in mathematics.
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Many methodological approaches, such as clinical interviews (Clement, 2000) and teaching experiments (Steffe & Thompson, 2000) are well-equipped for investigating the schemes and constructed knowledge of individuals (Glasersfeld, 1988). Sellers (2020) extended the teaching experiment to the exploratory teaching interview for when the researcher aims to influence student thinking but is not testing a pre-determined hypothesis. Numerous researchers have tasked subjects with reading proofs to probe their thinking (e.g., Dawkins & Zazkis, 2021). I used proof-texts during a teaching experiment which modeled a subject’s thinking. This approach was profoundly effective in revealing inconsistencies to the subject in her thinking.
Working in small groups can be a daunting activity for students, yet also provide an opportunity to promote peer interactions and build an inclusive community (Tanner, 2013). In this focus group study, we present how students (n = 43) perceive collaborative class projects in Calculus I at Foliage University, a pseudonymized private, predominantly-White, “very high research” university in an urban metropolitan area in the Northeastern United States. Findings show that participants who highly favored group projects saw these collaborative activities as opportunities: (1) to meet students who can be their friends outside of class, (2) to practice Calculus in a different way, such as being a teacher to a groupmate to explain a concept, and (3) to apply Calculus in practical ways. Meanwhile, other participants disliked group projects because of (1) group dynamics, (2) logistical limitations, and (3) resistance to new ways of learning Math. These findings inform the next iteration of Calculus I group projects at Foliage University. The curriculum team will improve collaborative projects by (1) providing multiple reminders of the already existing pointed instructions and rubrics in multiple modalities, (2) providing best practices in group communication, in regards to norms for participation, work distribution, etc., and (3) providing clear connection to lessons for genuine buy-in from resistant students. In conclusion, collaborative group projects provide opportunities for improving peer social cohesion and practical knowledge of Calculus but need targeted improvements in implementation and genuine buy-in from potentially resistant students.
The study investigates STEM course patterns among Native American students transitioning from Chief Dull Knife College (CDKC) to four-year institutions. With Native Americans representing a significant segment of Montana's demographic, especially within higher education, there is a clear need to understand their experiences in STEM disciplines. A dataset merging student records from CDKC with the Montana university system from 2001 to 2019 will be analyzed to discern enrollment trends and academic outcomes, particularly in STEM courses.
This research brief will share the quantitative results of a mixed methods study that explores teaching methods and practices that are known and/or are being utilized by faculty of introductory college-level mathematics courses in the U.S. Using a cluster-random sampling approach, institutions were selected, and department chairs were contacted to forward the survey. The survey contained two inventories and demographic questions. Overall, 113 participants from 37 different states completed the survey. One critical finding was that participants who were members of a professional mathematics organization were more likely to utilize more methods and practices than those who were not members.
A growing body of research points to the transformative potential of engaging students as partners with faculty to humanize math education (Cook-Sather et al., 2023). To help liberate marginalized populations, a larger research study was developed to investigate how the use of Networked Improvement Communities (NIC) composed of key mathematics stakeholders could work to address issues of Diversity, Equity, and Inclusion (DEI) within introductory mathematics courses through critically transformative participatory action research. An empirically-rich and unique single case-study emerged from a NIC that recruited students in addition to the faculty as NIC members. We address the following research questions: (a) In what ways was student voice prioritized in the Networked Improvement Community? (b) How does the integration of students impact the power dynamics within the Networked Improvement Community? To answer these questions, we conducted a thematic analysis of structured observational field notes of the NIC meetings, semi-structured interviews of NIC members conducted in May 2023, and four reflexive journal entries completed by each NIC member. This process generated cross-cutting themes relevant to the NIC’s inclusion of student voice and perceptions of power. Findings from this study suggest that although the structure of the NIC was intended to uplift student voices and create a space where students and faculty were equal partners, outside power structures prevented students from fully viewing themselves as partners with faculty members.
Over the academic year of 2022-2023, seven Pennsylvania State University faculty from the departments of chemistry, math, physics, and statistics, have implemented standards-based grading methods (SBG) in a dozen classes, ranging from introductory to upper division and small (<25 students) to large (~500 students). We present results of an ongoing retrospective qualitative study that includes student interviews and focus groups. As we work towards building a grounded theory of students’ experiences with SBG, we have identified several preliminary emergent themes. First, many participants feel that SBG leads to deeper learning compared to traditional grading system. Second, students often highlighted the flexibility of SBG systems, particularly the common retake policy on proficiency checks or quizzes. Some students discussed using this flexibility to have more control over their learning, setting up the pace of their study, increasing motivation and reducing stress. One student mentioned difficulties that arose from such flexibility as they had to manage procrastination and work-avoidance behaviors. One focus group discussed the stress they felt related to the slow pace of progress and/or exhaustion from the large number of quizzes. Almost all participants reported some level of initial confusion with how SBG worked. Finally, some participants expressed mistrust towards the attribution of grades in traditional grading systems. These students conveyed that SBG grades felt more reflective of their actual understanding of the material.
Mathematics holds crucial importance for STEM majors. However, many of these students are often seen leaving STEM majors following their initial two years of college math courses. Research indicates that fostering a sense of belonging in mathematics can positively influence students' persistence in STEM fields. Summer bridge programs, designed to facilitate students' integration into STEM at the college level, serve as one such initiative to enhance STEM diversity and retention. This study examines potential shifts in participants' identities, sense of belonging, and confidence in mathematics from the beginning to the end of a summer bridge program. Surveys on mathematics belonging and identity were administered to bridge participants, and the data from interviews and surveys were analyzed to investigate changes in their sense of belonging and identity in mathematics throughout the program.
The study of modeling and dynamical systems is becoming increasingly important in the life sciences. The concepts of time series and trajectory are central to understand the behavior of a dynamical system described by differential equations. A critical skill is to be able to navigate between these two types of representations as they provide complementary information. In this exploratory study, I use the covariational reasoning framework to analyze to what extent students who have taken an undergraduate course on modeling have mastered the skill of sketching the trajectory associated with a time series, and vice versa. Analysis of interviews show that students do not systematically exhibit the same level of covariational reasoning when completing these related tasks. I discuss the influence of teaching practices.
In recent years both the teaching and learning of introductory mathematics courses, such as Precalculus, have made immense strides to improve students’ understanding of foundational skills such as covariational reasoning (e.g., Thompson & Carlson, 2017). Despite the emphasis being placed on teaching from a covariational lens, exponential functions continue to be taught from a correspondence lens (Ellis et al., 2016). In other words, the focus is on solving an algebraic expression that defines a static relationship between two variables instead of the underlying covariational relationship between two quantities. This work is a subset of a larger project to support students’ covariational reasoning and specifically focuses on exponential functions. We are guided by the following questions: What type of covariational reasoning is present in students’ responses on exponential growth/decay questions? How do the students’ responses align (or not) with the mathematical goals of each lab?
This poster presents preliminary efforts in a research and development project exploring a professional community and its work with assessment strategies grounded in the values and practices of active learning. The project’s aim is to create, refine, and illustrate a definition for such active assessment.Throughout the academic year 2023-2024, members of our professional community are implementing a variety of active assessment strategies, including variations on standards-based grading, group projects and portfolios, in a broad range of courses, from first-year general education courses through upper division electives for mathematics majors. This poster aims to share the motivation, definition, and initial thoughts on the implementation of active assessment, and present some challenges and questions that have arisen.
The purpose of this poster presentation is to highlight the variations in student thinking when working with difference expressions in the Cartesian plane — a fundamental skill crucial for grasping numerous Calculus concepts. Despite its pivotal role, previous research has shown students often struggle to conceptualize and express distances. We surveyed n=169 undergraduate math students to evaluate their ability to represent distances within the Cartesian plane using expressions in terms of both x and y. Through analysis of the responses, we identified two crucial components for completing the tasks: a) using a magnitude interpretation of subtraction and b) establishing the Cartesian connection. We found a significant proportion of students did not use either of these components appropriately, leading them to incorrect responses. Our findings show a need for instruction on the skills foundational to understanding distances in the Cartesian plane within Calculus education.
In this poster, I present a metasummary of articles employing graph-theoretic methods (i.e., Graphs, Network Analysis, Maps, etc.) published in proceedings from North American (SIGMAA-RUME) and European (CERME) mathematics education conferences from 2015 to 2022. I use the results of this metasummary to describe international trends in the choice of method and prominent research topic themes in mathematics education research amongst conference proceedings articles employing graph-theoretic methods.
Engagement is an important contributor to college students’ success. In recent efforts to improve instruction in gatekeeper courses like college Precalculus, mathematics instructors across the US are striving to engage all learners by implementing various forms of active learning in their classrooms. However, it is not always clear how students themselves experience the phenomenon of mathematics engagement. This poster presents a qualitative approach for inquiring into college students’ lived experiences of engagement in their Precalculus course using a graphing elicitation tool together with interviewing methods of hermeneutic phenomenology. During the poster session, conference attendees will be able to view sample participant graphs, learn about the benefits of the method, and discuss insights into Precalculus students’ mathematics engagement. Although mathematics engagement is a complex dynamic construct, the proposed research method offers an approach that can paint a more nuanced picture of the phenomenon, while simultaneously bringing forth students’ voices and experiences.
To address a documented need for professional learning opportunities for mathematics faculty, a recent project designed and implemented a series of online learning modules for faculty new to teaching mathematics courses for future elementary school teachers. The modules constitute an asynchronous short-course for faculty. Short-course designers had the role of Providers whose presence was implicit in the asynchronous course. In this poster, we explore short-course design and revision spurred by the distinctions in goals, resources, and orientations of both the target audience of faculty who were novices in teaching future teachers and the Provider-designers. Based on analysis of designer memos, faculty surveys, and faculty interviews, the poster focuses on how differences shaped redesign of the short-course after the initial pilot. The results differentiate among what constitute the professional skills needed by an instructor and professional skills as a teacher educator, the role of teacher educator skills in being consumers and selectors of activities (rather than producers or designers of them), and the still understudied forms of motivation behind the work of novice teacher educators.
This poster presents salient insights into an inquiry-oriented transition-to-proof course, focusing on enhancing undergraduate students’ set-based reasoning for mathematical proofs. We highlight the distinct aspects of this course in building a learning community, effectively delivering instructional strategies, employing formative assessment, and addressing student challenges.
Sense of belonging is a critical factor in supporting female minoritized students’ persistence, motivation, and positive outcomes in STEM education. However, current research base is largely quantitative, focused on singular identities, and situated in the campus or department level, rather than the classroom level, at Predominantly White Institutions (PWI), Historically Black Colleges and Universities (HBCU), or selective research universities. This study addresses this gap by examining belonginess at the classroom level by implementing a sequential, explanatory mixed methodology to explore Black and Latina female students’ sense of belonging at a diverse open-access, minority-serving, public college. The poster presentation will include quantitative results and graphs that show how sense of belonging changes through a semester of college algebra and precalculus classes based on students’ racial and gender identities as well as their affinity for mathematics.
A General Education Data Literacy and Visualization Course Using Service Learning
In this poster, we describe an emerging digital task design principle: That students’ attention can be directed toward theoretically salient aspects of a representation by making those aspects of the representation interactable. We use Paoletti et al.’s (2023) framework to determine what aspects of the representation are theoretically salient and to analyze students’ activity. We present data to illustrate iterative cycles of task design, fine-grained analysis of students' activity, and revision according to our emerging task design principle. We highlight the efficacy of each revised task and provide implications for teachers and researchers as they adapt or design digital graphing tasks.
With increasing linguistic diversity in college mathematics courses, there is a pressing need to better understand the learning experiences of multilingual students. The work presented in this poster stems from data collected in a larger study about the experiences of 28 multilingual students in introductory mathematics courses. In this poster, we present narratives from six students using poetic transcription analysis. These narratives explore the role of “silence”, presenting experiences of students being silenced and also using silence as a tool to navigate the sociopolitical contexts of the classroom.
Proportional reasoning is foundational to understanding and communicating about issues of equity, fairness, and (in)justice. Yet, many students learn about mathematics and proportional reasoning in decontextualized ways, obscuring the role that mathematics can play in examining and impacting issues of social justice. We developed a lesson for a synchronous online university senior capstone course in which students explored claims in newspaper articles about the election of Indigenous Congresspeople through making sense of proportions. Our research question is to what extent do students see mathematics as connected to social justice issues after the lesson? Students did connect their mathematical learning to social justice, recognizing that mathematics can help them make sense of the world and advocate for change.
We describe the design and implementation of Cafecito con Matemática, a bilingual mathematics outreach event held at local bilingual elementary schools that serves predominantly Hispanic/Latino communities. During this event, we engage students and their families in hands-on mathematics games and activities that draw inspiration from Hispano cultures, communities, and traditions. We recruit, as teaching assistants for this event, mathematics graduate students and pre-service teachers who speak Spanish as their home language. Discussions with students and their families, with school teachers and administrators, and with the mathematics graduate students and pre-service teachers highlight the different perspectives and values that emerge from participating in this bilingual mathematics outreach event.
Our goal in this paper is to explore how beginning teachers transition between the figured worlds of dialogic and direct instruction, and how their current and desired teaching identities evolve in the process. We pursue our goal by focusing on a longitudinal case study of Olive, a beginning mathematics teacher, whom we followed for three years: as a pre-service teacher (PST), an intern (INT), and a novice teacher (NT) in her own classroom. Our findings show that as PST, Olive’s current and desired identities were coherent and aligned with the dialogic instruction figured world. This desired identity remained dialogic throughout the internship, even when she taught her own class. However, as a novice INT, Olive had to align her teaching actions with those of her cooperating teacher. Olive’s discourse reflected a tension between her current direct teaching and her desired dialogic identity. Olive continued experimenting with dialogic teaching in her own class (while still as an INT), still holding it as desirable (“letting them do the talking rather than me”). However, her teaching comprised a mix of dialogic and direct actions, which created another gap within Olive’s current identity. As a first year NT, the conflicting identities moved towards reconciling, in an interesting way. Olives’ current and desired identities differentiated: dialogic for “advanced” students and direct for “non-advanced” classes. Interestingly, Olive did not seem to be aware of potential inequities this reconciliation identities may introduce into her teaching. Our study contributes to a better understanding of the beginning mathematics teacher trajectory between two figured worlds, the specific junctions along this trajectory where identity conflicts may arise, and the types of conflicts. The findings suggest that beginning teachers may need support in negotiating their dialogic vs. direct instruction alignments beyond supervised internship into the early years of autonomous practice.
The news media is the primary source of information about social and scientific issues for most adults (Bissonnette et al., 2021). Statistical and mathematical products in the news take on many forms, requiring inter-related skills to appropriately comprehend and evaluate (Gal & Geiger, 2022). The purpose of this qualitative pilot study was to investigate how students in an introductory statistics course utilizing a simulation-based curriculum apply statistical concepts when critiquing statistics in the news media. Task-based interviews involving mock news headlines and accompanying articles were used to examine students’ thinking using Watson and Callingham’s (2003) six-level construct of statistical literacy. Ongoing analysis includes identification of themes in students’ responses. Ideas for future research directions, adjustments to tasks, and pedagogical recommendations will be discussed.
Calculus concepts require students to express and interpret distances in the Cartesian plane. Yet previous research shows that many students are not easily able to do so. To effectively represent distances, students must be able to understand and apply the Cartesian connection. The Cartesian connection is understanding that a point is on graph A if and only if its coordinates (x, y) satisfy the equation of A. Students should also be able to flexibly move between algebraic and graphical concepts to represent distances both in terms of x and y to determine if points lie on a line. We are interested in why some students may not display this flexibility and what underlying issues are preventing students from making the Cartesian Connection.
Transitioning between university teacher preparation programs and secondary schools is a complex process, resulting in many of the ambitious teaching practices experienced in teacher preparation programs being absent, but may reemerge later. This study explores how three beginning teachers negotiate between their formal education and the realities of classroom practice, specifically concentrating on the integration of reasoning and proving into their teaching. We use Herbst and Chazan’s (2003) theory of practical rationality to conceptualize this transition. Our findings suggest that initially, the participants discourse expressed a need to adhere to school rules, curriculum, and the cooperating teacher’s traditional practice. With more experience, the teachers discussed their desire to implement more activities that require justifying, conjecturing, and exploring. Once the participants became novice teachers and gained their own classroom, their obligations shifted to the individual student. Despite the added responsibilities, the beginning teachers continued to incorporate reasoning and proof into their classrooms. These findings shed light on the process of reconciling various teacher obligations while maintaining some focus on reasoning and proof.
Exploring the impact of Large Language Models, our study investigates how AI agents facilitate cross-cultural collaboration in undergraduate mathematics. Precalculus students from a U.S. university collaborate online with calculus students in China, solving mathematical tasks. We anticipate that AI will encourage communication across cultures while enhancing problem-solving skills. Thematic coding of written work, surveys, and Zoom recordings explores the AI assistant's role. Ethical considerations guide our approach. The poster presentation will illustrate the study's design and share anticipated cross-cultural insights.
Learning assistants (LAs) are undergraduate near-peer tutors who, having previously been successful in some course, aid in the instruction of that course, typically by interacting with individual students or groups. It has been shown that LAs’ presence in classrooms is associated with positive student outcomes, but there is little understanding of how LAs contribute to such outcomes. This study contributes a subject-specific perspective on LA classroom practice by examining discourse between LAs and students on the topic of implicit differentiation in a Calculus I course. Initial analysis indicates that LAs aid students in learning this topic by coordinating across multiple representations of implicit differentiation and integrating previous calculus content knowledge into these components. The study has the potential to bolster our understanding of student learning on the topic of implicit differentiation and gain insight into LAs’ instructional moves as they interact with students.
Supporting faculty to transform their views of self, students, and authority can begin with the course syllabus. This report gives results of analysis of 39 mathematics syllabi based on Taylor and colleagues’ (2019) Social Justice Syllabus Design Tool. The tool provides reflective questions in three categories: course community, relationships, and processes. The poster will include visual data summaries and comparisons of the degree to which the key constructs occurred in syllabi for 100-level, 200-level, and 300-level-or-higher courses. Additionally, the poster will offer examples of the language used in syllabi that communicated messages that addressed teaching and learning mathematics with, about, and for justice.
This study sought to investigate the experiences of international and non-international students in chavrusa-style mathematics courses (MathChavrusa). Chavrusa-style learning involves the long-term pairing of students, with classroom activities centered around constructing understanding from learning materials through debate. Interview and class observation data were collected from international and non-international students taking graduate mathematics classes. Findings indicated that MathChavrusa provided both international and non-international students easily accessible sources of support when grappling with new material. Social and schedule compatibility of partners were noted as important considerations for both groups of participants. A benefit noted by international student participants was that long-term partnerships allowed them to more easily build relationships that extended beyond the classroom. Both participant groups also noted that the mode of instruction (i.e., in-person, hybrid, and online learning) influenced the level of collaboration and socialization present in their MathChavrusa experience.
Undergraduate mathematics features a significant presence of Asian students, whether international or American (NCES). These students naturally bring their views on learning mathematics from their cultural experiences. However, researchers often adopt theories of learning and teaching that may not fully incorporate the students’ viewpoints when developing instructional materials and analyzing student reasoning. This proposal aims to foreground Asian scholarship by reviewing existing literature in this regard and intends to open a venue for discussing ways the RUME community could begin addressing this issue.
Based on research in the field of philosophy education, argument mapping has the potential to improve student comprehension of proofs. For this poster, I will report a literature review of the current use of argument diagramming in mathematics education and in adjacent fields (like philosophy and logic) and present a study design which seeks to address the question: how does diagramming the argument of a proof impact a student’s comprehension of that proof?
The Program Assessment Conference in Mathematics (PAC-Math, DUE# 2036211) was hosted in October 2023 at the University of Tennessee Knoxville. The conference implemented a distributed leadership model in the context of a networked improvement community. This structure provided conference participants with the opportunity to take on leadership roles and actively define the direction of group discussions. This poster will share insights gained from PAC-Math into how distributed leadership can support a networked improvement community.
The Program Assessment Conference in Mathematics (PAC-Math, DUE# 2036211) was hosted in October 2023 at the University of Tennessee Knoxville. The conference produced key components to build a protocol for mathematics departments to self-assess their professional development for teaching provided to graduate students in mathematics. We describe previously presented results of a census survey of mathematics PhD programs conducted in Fall 2021 that reinforces the need in higher-ed mathematics for tools to do this assessment work, confirming results from a similar survey conducted in 2015. In this poster we outline the protocol, departmental resources needed to undertake this assessment work and questions of interest for the RUME community.
Over the last two decades, the field of biology has become increasingly more quantitative leading to more mathematics being integrated into biology education curriculum. However, students enrolled in biology courses often report having negative feelings towards mathematics. To understand these negative feelings, this study used cluster analysis to identify patterns of mathematics motivation among undergraduate students enrolled in a gateway biology course at a large research-intensive university. For data, we used a survey, derived from Expectancy Value Theory (Eccles et al., 1983), students completed at the beginning of the course that investigated six affective variables: expectancy of success, utility value, intrinsic value, mathematics attainment value, effort cost, and emotional cost. The findings show that students in the introductory biology course can be clustered into three motivational groups (low, moderate, and high), with over half in the moderate motivation cluster. For future analysis, we plan to further explore these groups by investigating gender and pre-professional status in each group.
In this mixed-methods study we characterize the physics quantitative literacy (PQL) progression of physics majors at a large, public university. We combine quantitative data, as measured by the Physics Inventory of Quantitative Literacy (PIQL), and qualitative data from a free-response survey item that captures student reflections on the fact that PQL is slow to develop for physics majors. We share preliminary results that show students with comparatively high PIQL scores were likely to express surprise that PIQL results do not typically improve over the course of instruction. We also found that lower scoring students were likely to express self-doubt and a desire for instruction that addresses quantitative literacy as measured by the PIQL. This work is a first step in a larger, ongoing project that contributes to the understanding of how students' PQL develops across the physics major, and how instruction can help all students.
Constructing proofs and working with formal mathematics is an important feature of studying advanced mathematics courses. Topology is a particular subject whose proofs typically demand both formal and spatial perception. As the ability to transition across various modes and levels of thinking seems imperative in topology, a model that can help describe both the form and depth of thought would be of great value. We seek to develop such a model by networking Tall’s three worlds of mathematical thinking and APOS theory.
This poster will focus on challenges and benefits that have appeared in the first year of implementing active assessment. Active assessment is a term we use to describe assessment strategies grounded in the values and practices of active learning, including attention to equity. The authors are members of a community of practice that is part of a larger project which explores instructor and student perspectives on active assessment. We will share stories about instructor motivation as well as initial and in-progress thoughts on the implementation of active assessment, specifically keeping in mind ways that active assessments may be more or less equitable than timed, independent, written exams. We use these stories from our practice to frame some of our ongoing questions related to equity and the challenges and benefits of active assessment.
A professional role of mathematics graduate students is complex, as it entails being simultaneously a student, a teacher, and a researcher. While performing duties required by the three roles, an inquiry about the existence of relations between them arises. Hence, the goal of this poster is to build and discuss a model “Interrelation Between Graduate Students’ Teaching and Learning Mathematics”. Using Schoenfeld’s (2015) framework (ROGs), the proposed model suggests the impact of graduate students’ studying of advanced mathematics on their teaching, as they use their previously acquired knowledge to shape their decisions for lecturing, problems presented and solved, and responding to students’ questions. In addition, Ball et al.’s (1998) (PCK) framework is utilized to suspect an influence pedagogical content knowledge has on the way graduate students perceive and solve challenging mathematical problems.
The Physics Inventory of Quantitative Literacy (PIQL) is a 20-item multiple-choice test designed to measure the development of students’ physics quantitative literacy (PQL) across three facets: ratios and proportions, covariation, and signs and negativity. Repeated testing across multiple courses, coupled with requiring up to 40 minutes for students to complete the test, could lead to testing fatigue and unreliable results. We seek to create two shorter versions of the PIQL (a.k.a. piqlets) that are statistically equivalent to each other in terms of student performance on all three facets of PQL. We considered 240 combinations of items for the piqlet versions, subject to constraints that both the content and the format of items were equivalent across the two versions. Data were collected by administering the PIQL in three introductory physics courses at a large public university (N ~ 2500). Piqlet versions were compared by calculating differences in average scores, reliability coefficients, and psychometric parameters from item response theory (IRT). The combination of items that produces the most similar piqlet versions in our data have score differences less than 1.3%; Crobach's alpha values of approximately 0.8, with differences less than 0.01 between versions; and IRT parameters that differ by no more than 0.1 for items included on both versions.
Professional development (PD) workshops on teaching have been shown to effectively increase undergraduate math instructors’ use of research-based instructional strategies (Archie et al., 2022). However, face-to-face PD is not accessible for all math instructors, including those with young families, those at under-resourced institutions, and those concerned about the environmental impacts of travel. In response to these constraints, the Mathematical Association of America (MAA) sponsored a series of nine online PD workshops in summer 2022. The MAA solicited proposals and selected teams to lead workshops on a variety of mathematics teaching topics. The purpose of this preliminary research was to explore the outcomes of the 2022 workshops.
Statistical reasoning is inherently different from mathematical reasoning because reasoning about data involves reasoning with uncertainty. In mathematics, deduction and induction are privileged, but abductive reasoning is powerful when reasoning about data, especially when making inferences from sample data. I present preliminary results from examining the forms of reasoning—deductive, inductive, and abductive—that novice statistics students employed when reasoning about and with sampling distributions.
This poster has two purposes. First, we will describe some of the latest techniques of meta-analysis and meta-synthesis through the lens of a current project looking at algebraic teaching interventions for grades K-12. This includes illustrating the use of cutting-edge software for conducting a meta-synthesis such as Covidence and Raayan and looking at a new technique for extracting qualitative data from an article using artificial intelligence. Second, we will describe some of the challenges and opportunities that have resulted from this process to highlight how pre-service teachers and researchers could benefit. This includes discussions about the definition of an algebraic teaching intervention for screening, a look at some of the interventions that were discovered during search/screening that might be applicable for teaching, and a discussion about what information from an article would be useful for a practicing educator.
Existing studies about the fraction thinking of undergraduate developmental mathematics students seem to take a deficit perspective, focusing more on what students in this demographic do not understand about fractions rather than reporting what is productive in their thinking (Alexander, 2013; Baker et al., 2012; Doyle et al., 2015). In this poster, I will report on tasks that I developed to examine the fraction thinking that undergraduate developmental mathematics students can utilize. I drew from existing K–12 fraction literature when designing these tasks to be used during clinical interviews.
This study explores first-semester calculus students’ use of mathematical problem-solving (MPS) strategies while working related rates of change (RRC) problems in both an online homework format and a traditional pencil-paper format. We address two research questions: (1) How do students’ MPS strategies when working online RRC homework problems compare with their MPS strategies when working RRC homework problems in paper-pencil format? (2) What influence does the ‘view an example’ feature in online homework have on a student’s MPS strategies when working an online RRC homework problem? Using midterm performance levels on RRC problems to select participants, we then conducted task-based interviews in which participants were asked to solve four (two paper-pencil and two online) RRC problems. The findings suggest that students may be using features of online homework platforms on RRC problems in a manner that circumvents the use of MPS aspects of online homework platforms.
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There was a successful departmental effort to improve the calculus DFWI rates at a midwestern university (Author & Author, 2021). To maintain this change during the pandemic, instructors developed Desmos-based activities to encourage student engagement and active learning. This platform allows students to interact with tools on activities designed by their instructors. Since the implementation, instructors sought feedback from the students on the digital experience. This study focuses on a partnership between the students and the instructors to develop a rubric to evaluate Desmos activities for the Calculus course.
In recent years, student reasoning in linear algebra has been a rich area of research, but relatively few studies have focused on linearity and proof in this area (Stewart et al., 2019). In this study, we analyze students’ evoked concept images of linear transformations in the context of a Desmos module focused on transformations in the plane and identify aspects of the module that students reported finding helpful for their learning. This report draws on the written responses of 54 U.S. undergraduate linear algebra students at the end of a unit on linear transformations.
Physics education researchers have begun to uncover the different ways that mathematicians and physicists interact with symbols and do math in physics contexts. Recent work has identified an important expert behavior in physics problem-solving in which expert physicists, much like jazz musicians, riff with mathematical objects to work toward an answer, realize when a particular approach will be fruitful or not, and adjust accordingly. In this qualitative study, we characterize some resources that appear to support mathematical riffing in physics as part of an ongoing effort to better understand the role of mathematical intuition in physics. Our study consisted of think-aloud interviews with physics graduate students in which they were asked to solve an unfamiliar graduate-level physics problem, using familiar mathematics. In this poster, we identify a feature of physics mathematical riffing, which we call “mathematical impetus” that was consistent across graduate students and describe how it bolsters mathematical intuition.
Realistic Mathematics Education (RME) is a curricular design theory that involves designing materials with realistic starting points and guiding students through four levels of activity: situational, referential, general, and formal (Gravemeijer, 2020, 1999). This theory has been used to develop a multitude of curricular materials including ones that incorporate technology. This poster investigates the design of three different technologies (video games, GeoGebra, and applets) designed to elicit different forms of activity in an RME sequence. Each technology has a 2D and a 3D version and was either could be used or was designed to be used in a linear algebra classroom. This poster will present the design heuristics of each type of technology that lend themselves to a particular type of activity to be evoked with the goal of providing insight into designing technology for use in an RME-type sequence.
This study explores the experiences of two Ph.D. women mathematicians from groups historically disenfranchised in mathematics, Anna and Sasha, in their undergraduate and graduate real analysis courses. As such, we aim to better understand these experiences for students from groups historically disenfranchised in mathematics with an eye toward informing instructional practices in advanced mathematics courses that support the mathematical development and pathways to achieving a doctoral degree in mathematics. We analyze participant interviews that focused on questions about their undergraduate and graduate classroom environments, their recollections of important concepts, and the role of their experiences in the development of their mathematical identity. Using tenets from Critical Race Theory (CRT), we address the following research questions: (1) What belief structures about how participants think of themselves as mathematicians were evoked by their experiences? (2) To what extent did their courses or experiences in real analysis advance their development as mathematicians?
To better understand retention and attrition in the mathematics major, our research team launched the Mathematics Journeys of Retention: Why Individuals Shift Educational Paths (MAJORWISE) project. During October 2023, we carried out a nationwide survey in the United States with free-response items of students who have been enrolled in a mathematics major during the years 2013-2023. In addition to demographic information, this survey solicited information such as reasons for enrolling in a mathematics major and timing of and reasons for deciding to either leave or stay in the mathematics major. We analyzed free-response survey items using a predetermined coding scheme framed by Self-Determination Theory (Ryan & Deci, 2000), with codes for autonomy, relatedness, and competency. The findings from this project will provide insight into how relevant stakeholders can best promote the motivation and retention of mathematics majors.
The study reported here was conducted in spring 2020, during the COVID-19 pandemic, when the instruction changed from face-to-face to online. The change affected the interactions instructor-students-mathematical content. In this study, we compare online vs face-to-face instruction, with a focus on affordances and interactions. Participants (N=51) were undergraduate students enrolled in a Discrete Mathematics course taught by the author. The mathematical content referred to proofs by mathematical induction. Data collected included a questionnaire for students, a questionnaire for instructor, students’ postings in Blackboard/Discussion Forum (homework and comments on mathematical induction tasks), and students’ answers on the final exam on two tasks referring to mathematical induction. The findings revealed one important online affordance for students, the Blackboard/Discussion Forum. To account for the patterns of online interaction between students, social network analysis was employed.
We describe evidence of the usefulness of the Comparative Judgment approach to assess the quality of explanations in mathematics.
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Feedback is one of the more instructionally powerful and least understood features in instructional design (Cohen, 1985). Past studies offer inconsistent results on most features of feedback and little is known about how some of these factors benefit students’ interactions with feedback such as an instructor’s specific choices in the way their comments are phrased, the feedback’s potential intent, an instructor’s inclusion of praise, and the purposeful addition of questions that may allow students the insights to reconsider their initial response.
Proving trigonometric identities represents a major milestone in students’ development of proof concepts for undergraduate mathematics. However, few studies have characterized curricular approaches to trigonometric identities. The study presented here examined the ways in which five precalculus textbooks (1) invite students to derive trigonometric identities and (2) target a variety of reasoning and proving actions in trigonometric identity problems. Each research question corresponded to a set of codes that were applied to textbook content. The poster will highlight the major trends observed in the relative frequencies of codes across the five textbooks.
Students in undergraduate mathematics classes are using online resources to support their learning with increasing frequency (Author, 2019, 2020). The use of such online resources has not proven to be straightforward for students. Undergraduate students have well-documented difficulties locating, evaluating, and making use of information sources regardless of the topic area in question (Scott & O’Sullivan, 2005; Walraven, Brand-Gruwel, & Boshiuzen, 2008). These difficulties may vary depending on a student’s familiarity or comfort with the field (Brand-Gruwel, Wopereis, & Vermetten, 2005). Thus, it is important to develop subject-specific models of student information-seeking. This is particularly important for the field of mathematics due to the role that lower-division math classes have often played as a gatekeeper for entry into STEM majors (Martin, Gholson, & Leonard, 2010). This poster will present findings from a nationwide mixed method research project that has collected surveys from over 300 students across the country and follow-up semi-structured interviews with over 60 students intended to assess how students currently make use of online resources to support their learning in lower-division mathematics classes. We will report on how students describe the efficient and ethical use of online resources to study mathematics.