RUME 27
2025
Alexandria, Virginia
Number of pages: 1625
Number of papers: 294
- Samuel Cook
- Brian P Katz
- Kathleen Melhuish
2025
Alexandria, Virginia
Number of pages: 1625
Number of papers: 294
Algorithms, proofs, and definitions all play important roles in mathematics as products of research and objects worthy of study in the classroom. Nevertheless, limited research has examined mathematicians’ views of algorithms or compared their views across these objects and related activities. Based on interviews with nine discrete or computational mathematicians, we highlight similarities and differences between the values associated with these mathematical activities and relative importance placed on the varied activities.
Teaching is a complex process in which mathematicians are expected to make many decisions before, during, and after the instructional time. These decisions can be shaped by various factors such as pedagogical or content knowledge, professional obligations, or the personal beliefs and values of the mathematician. In this study, we draw on the theory of Practical Rationality as a means to examine the role that professional obligations play in the pedagogical decisions that mathematicians make. Specifically, we examine the role that individual obligations play in instruction noting that the way in which the obligation is viewed can lead to different decisions
In this report, we use the Schema portion of Action-Process-Object-Schema (APOS) theory to frame a study of student understanding of the differential calculus of two-variable functions. The use of Schemas allows us to obtain a more holistic perspective of students’ understanding. Data was obtained from semi-structured interviews with eleven students. We argue that the APO portion of the theory can give detailed information about specific Schema components but that it does not inform about the interrelations between the different components. However, the notion of Schema and its instrumentalization via the notions of genetic decomposition, types of relations between components, and the triad stages of Schema development can provide a more comprehensive picture encompassing the different components of the differential calculus Schema. The notions of the scope of an instrument and the graph of a Schema are introduced to afford a visual representation of a student’s Schema.
Researchers have investigated several interpretations of definite integrals and their notation, and they have recognized that one particular interpretation, Adding Up Pieces (e.g., Jones, 2013), is especially important for applying definite integrals to physics and other domains. This same research has paid less attention to how students partition physical attributes, and thus construct pieces, in ways consistent with constructing Riemann sum approximations and definite integrals. We analyzed how 8 students taking calculus-based physics courses constructed Riemann sum approximations and integrals for one task about water pressure on a tank wall. All of the students recognized that integration was relevant, but half did not partition the wall in ways compatible with constructing Riemann sum approximations or definite integrals that solved the task. We conjecture that seeing integrands as rates of change and structuring situations in terms of level curves for rates of change is a significant, but underexamined competency.
The ability to make a logical inference is at the heart of mathematical experience. However, a personal reasoning does not always follow the rules of formal logic. In this case study we focus on responses of one participant, a secondary school mathematics teacher, to scenarios involving coercive logic (the construct is defined and exemplified in the paper). This allows us to take a detailed look into what guides and what influences the participant’s approaches to various scenarios. We argue for the relevance of coercive logic to mathematics education, in particular, in exploring mathematical reasoning of students.
In this study, we consider students’ performance on tasks involving solids of revolution to refine a recently introduced, research-based model. An activity was designed with tasks to help students do the constructions proposed in the model. After implementing the activity in the classroom, we suggest additions to the model using data from the written responses of 22 university calculus students’ quiz and exam tasks. The results provide information about the challenges that students face when dealing with solids of revolution and allow us to refine the model to add more detail that might be used to help students overcome observed challenges. We discuss some limitations of the study and share teaching implications.
Research on students’ construction and interpretation of integrals has grown significantly over the past decade. The present paper contributes to recent research on challenges which students experience while coordinating the “product layer”, 𝑓(𝑥) ∙ 𝑑𝑥, and the definite integral in applied contexts. In particular, we demonstrate a case study which examines how an undergraduate student, Matthew, linked the “product layer” and definite integral in an introductory physics task by blending mathematics, physics, and visualization knowledge resources. We analyzed Matthew’s moment-to-moment reasoning by utilizing a modified dynamic conceptual blending diagram (MDCBD), coming from the dynamic conceptual blending perspective, and an expanded Visualization/Analysis/Physics (VAP)-model, as detailed in (Sus & Izsák, 2024). Our findings indicate that students’ understanding of multiplication with quantities in applied integration contexts is highly complex and that neglecting the dynamic interactions of knowledge resources represents a significant gap in integration research
Many instructors observe that students often rely on memorization in introductory college mathematics. However, existing literature on effective teaching strategies does not fully address the need to cultivate both conceptual understanding and deeper orientations towards understanding in students’ learning. This study presents a case analysis of calculus lectures delivered by an experienced instructor at an R1 institution, revealing significant opportunities for enhancing conceptual learning and promoting stronger orientations towards understanding among students. The analysis identified four key strategies employed by the instructor: 1) framing instructional activities as a pursuit of understanding; 2) examining underlying meanings while connecting students’ intuition; 3) providing motivations, justifications, and multiple perspectives; and 4) creating cognitively challenging tasks and encouraging students to embrace discomfort in learning. These simple, practical strategies do not impose significant demands on instructors yet hold great potential for helping more students approach mathematics through deep understanding.
This research investigates significant moments in college students' drawn life stories in mathematics with the aim of examining implications for STEM engagement and potential reasons for gender-related differences in participation in mathematics intensive STEM fields. My research employs drawn life stories in mathematics specifically focused on students’ experiences with mathematics to understand students' evolving relationships with mathematics, encompassing positive and negative experiences. Employing mixed methods analysis of visual and written mathematical narratives combined with qualitative interviews, my research explores the effectiveness of allowing participants to visualize their mathematical journeys, and quantitatively assesses the impact of recurring themes in students’ drawn life stories in mathematics. This research advances our understanding of factors that influence the nature of students' relationships with mathematics, and potentially inform strategies for enhancing STEM participation and gender inclusivity, using a unique approach in mathematical narrative research.
As the push for active learning pedagogies in undergraduate mathematics continues, research into the teaching practices of graduate student instructors (GSIs) has expanded significantly. This paper presents findings from a multiple-case, descriptive case study of the demonstrated mathematical knowledge for teaching (MKT) of GSIs within the context of active learning Precalculus lessons. This study aims to bridge the theoretical and practical distinctions between the MKT knowledge domains as they are used by GSIs in their teaching practice. Practical implications for the professional development offered to GSIs, as well as theoretical implications for studying the MKT construct in post-secondary settings, are discussed.
This study investigates how students at a Southern California Hispanic Serving Institution (SCHSI) perceive and define smartness in the mathematics classroom. We employ a qualitative approach, analyzing narratives from an open-response survey question that asked students, "What do you think it means to be smart at [SCHSI] in math?" The findings challenge the traditional binary categorization of dominant and counter narratives, revealing a more nuanced understanding of smartness. The analysis identifies a spectrum of student perspectives, ranging from those emphasizing performance outcomes (e.g., grades, passing) to those prioritizing process-oriented approaches (e.g., understanding, effort, seeking help). The results underscore the importance of recognizing and valuing diverse forms of smartness in mathematics education, particularly within community college settings serving underrepresented student populations. By fostering an inclusive learning environment that acknowledges and supports various conceptions of smartness, educators can empower students to develop a positive mathematical identity.
Algorithms play increasingly important roles in mathematics as they facilitate solving complex problems both in teaching and research contexts. Nevertheless, limited research has examined mathematicians’ views of algorithms or the broader goals of the mathematical community that relate to algorithms or work therewith. Based on interviews with nine discrete or computational mathematicians, we highlight three values upheld by a variety of norms related to algorithms. Of note, some of these values and norms appear to be in tension.
As the uses of data in society continue to expand, it is vital to prepare students to engage with data in meaningful and critical ways. This study used a course-wide assessment in an introductory statistics course to explore undergraduate students’ statistical literacy in media contexts based on Watson and Callingham’s (2003) six-level construct. Statistical analysis found student performance on non-media tasks had limited association with performance on media- based tasks. Students demonstrated higher levels of statistical literacy when provided with specific prompts. Findings suggest the need to develop students’ propensity for critical questioning when engaging with statistics outside classroom settings through intentional incorporation and discussion of media items in introductory statistics courses.
Student learning of physics is built on a foundation of mathematical knowledge. Accordingly, physics education research has devoted much study to students’ use of specific mathematical content in specific physics contexts. More generally, though, students also experience various “cultural” differences between the (mathematics) courses where they learn this content and the (physics) courses where they apply it. Researchers have discussed these cultural differences, but to our knowledge they have not investigated whether students are aware of them, or students’ beliefs about how these differences may affect their learning. We used a survey questionnaire and individual interviews to study this in a group of math-physics double majors (or majors in one with minors in the other). These students are indeed aware of many such cultural differences, cite many classroom examples, and have strategies to cope with them.
Lockwood (2014) articulated the importance of sets of outcomes, suggesting that a “set-oriented perspective” is a productive way of thinking for students as they solve counting problems. While this construct has indeed proven useful, it remains quite broad. In this report we aim to provide more detailed insight into the variety of ways students engage with outcomes while counting, exploring ways that we might elicit robust set-oriented perspectives in students. We present data from student interviews that were designed to elicit engagement with sets of outcomes, and we report on a variety of ways in which students worked with outcomes as they solved problems. We highlight contrasting cases that emerged in these interviews, making the case that there is room for nuance (and differences) in students’ engagement with sets of outcomes in counting.
We analyze the pedagogical moves related to analogies used by a quantum mechanics instructor to support a class community in developing a shared unified understanding of eigenequations across the contexts of linear algebra, spin, energy, and position. We characterize an instructor’s pedagogical moves as he engaged students in analogical reasoning. Some moves include posing tasks conducive to analogizing; preparing, soliciting, and scaffolding students’ participation in analogical reasoning; using deictic gestures and inscriptions; juxtaposing symbols representing the analogized concepts; and explicitly highlighting the sameness of the analogized concepts. We exemplify these pedagogical moves with analytical descriptions of illustrative class episodes. We discuss how these moves can support the class community’s expansion of their common ground by fostering the development of the class’s shared unified understandings of eigenequations.
Prospective teachers need hands-on experience in the field before they can independently manage their own classrooms. Although experiential learning is widely acknowledged for its benefits, its implementation in content-focused mathematics courses for prospective K-12 teachers remains limited. To address this gap, mathematics content courses for future teachers should offer experiences that extend beyond traditional classroom settings. One effective approach to achieving this is through incorporating service learning experiences. This paper explores the integration of service learning into three distinct mathematics courses for prospective K-12 teachers, including two upper division mathematics courses, highlighting how this approach was implemented in each course. Results suggest service learning in upper- division mathematics courses benefits prospective teachers by fostering experiential learning, enhancing collaboration, communication, and career skills, as well as deepening engagement with course content and the community.
Using narrative analysis on experiences from Andy, a mathematics graduate student instructor, teaching for the first time, I address the question: How do frames of teaching and learning compare before, during, and after teaching for the first time as the main instructor? I focus on Andy’s frames while reconstructing her semester-long experience, attending to the events that shaped her perception of these frames. For one, her “business” frame of teaching that she entered the semester with was at odds with her newfound responsibilities as the content instructor. This and other findings discussed have implications for the professional development and mentoring of future mathematics content instructors, including how newer instructors might be better oriented to manage and attend to classroom events.
Sociopolitical lenses applied to frames research have affirmed that frames are culturally constructed. A subsequent curiosity, then, is to uncover what aspects of culture and context shape an instructor’s frames with respect to teaching and learning mathematics. To this end, this study followed three mathematics graduate student instructors during their semester of teaching for the first time. Through a grounded theory design, informed by critical theories, the author found themes within beliefs, values, and norms were salient in these teaching experiences. One consequence of this study is giving names to the aspects of culture and context which shape frames. This insight is valuable for understanding how to shape asset-based frames.
Mathematics defines the propositional conditional ‘if P then Q’ as the truth-functional material conditional: it is false if P is true and Q is false, and true otherwise. This is counterintuitive with respect to natural language in that ‘missing-link’ conditionals (with true but unconnected P and Q) are considered true, as are conditionals with false antecedents. How is this reflected in the examples used by introduction-to-proof textbooks? Do these use natural-language conditionals in their explanations, or avoid them in favour of mathematical conditionals? What characteristics do their chosen examples share, and where do they diverge? We address these questions using a theoretically informed qualitative analysis of content on conditionals in 17 commonly recommended introduction-to-proof textbooks. We consider advantages and disadvantages of different approaches in relation to research in philosophy and psychology as well as in undergraduate mathematics education.
We take a situated cognitive perspective to better understand how undergraduate students utilize study resources in their introduction to proofs (ITP) course, particularly compared to their use of these resources in pre-proof courses. Prior research has shown that the transition to proof contains an increase in expectations of rigor compared to previous undergraduate mathematics courses. However, less is known about how students select particular study resources to meet these new expectations. We conducted task-based interviews with four undergraduate students enrolled in an ITP course to better understand the study resources they employed in both pre- proof and proof-based contexts, and the potential motivating factors behind adjustments to study methods. We found that students attempted to migrate previously used resources as well as adopting new resources and were motivated to adjust some study habits due to the novelty of the ITP context, a lack of relevant resources, and instructor modeling.
Mathematics departments nationwide are tasked with addressing racial inequities, including students’ racialized experiences in calculus courses. Faculty must interpret racial information critically to advance equity through change initiatives. However, mathematicians often hold colorblind views of instruction that are reinforced by mathematics epistemological values, such as neutrality and objectivity. It is, thus, unclear whether mathematics epistemology inhibits the advancement of racial equity within mathematics departments. This analysis reports on racial sensemaking among 22 mathematicians at a research-intensive university amidst a reform effort to improve Black and Latin* students’ calculus experiences. Guided by theories of epistemic culture, I compare mathematicians’ mathematical interpretations of racial phenomena with those proposed by critical race theory. Findings exhibit how mathematicians conflated contextualizing racial information with stereotyping, and focused on quantities without recognition of students’ lived experiences or the social justice implications of their arguments. Implications are provided for using epistemology as a lever for change.
In this paper, we discuss our experience collaborating with mathematicians to increase their use of active learning pedagogy in a proof-based linear algebra course. We use this experience to attend to three primary research objectives. First, we identified three primary categories of instructor considerations that would determine whether or not they would incorporate a proposed strategy. Second, we observed and made sense of which of these were most prominent for these mathematicians. Third, we determined what combination of considerations needed to be satisfied to warrant the implementation of a strategy by these mathematicians.
Technology has become an integral part of undergraduate mathematics, particularly the use of technology to solve problems (i.e., the use of computation). In probability and statistics, this push has resulted in several projects designing and assessing tools that are conjectured to be advantageous to students and their learning. Despite this trend, minimal research exists on how students perceive the use of computational tools in their courses. As such, we designed a brief survey for students enrolled in introductory probability and statistics at a university in the Northeastern United States. Using thematic analysis, we qualitatively analyzed these survey responses to explore their perceptions of the integration of computation into their courses. Three themes were identified, relating to features of tools, augmentation of actions, and long-term benefits. This exploration of students’ perceptions allows us to better understand their views on computation and the need for professors to make instructional goals explicit.
Bilingual students are an integral part of the mathematical graduate education system, so understanding their perspective on learning mathematics is important to ensure they are provided proper support. While many students learn undergraduate mathematics in English, some graduate students’ first exposure happens at the upper undergraduate or graduate level, introducing additional challenges. The purpose of this study is to discover reasons for bilingual graduate students’ language switching when understanding and experiencing mathematics. All four bilingual participants were in a math PhD program and acquired their undergraduate degrees outside of the United States in their respective languages. The results revealed that students switch languages not only between reading textbooks or during lectures, but also when working on solving mathematical problems and internalizing difficult concepts.
An important goal in research on mathematical modeling is to understand the influence of teacher scaffolding moves on students’ model construction. This paper validates the use case of two common frameworks (one for scaffolding and one for student modeling processes) with STEM undergraduates. Study I investigated whether variability in scaffolding strategies could predict variability in participants' modeling processes. Findings suggested that the initial assumption—that scaffolding moves articulated by the framework are clearly aligned with specific modeling processes—was not supported. Study II further explored how scaffolding moves related to distinct stages of model construction and sought to improve cohesion of the categories of scaffolding moves. Our results challenge simplistic expectations about the relationship between scaffolding strategies and learner engagement, highlighting the need for more nuanced approaches to using these frameworks. The studies also demonstrate the importance of qualitative follow-up to refine our understanding of how scaffolding supports complex model construction processes.
Course coordination in high enrollment undergraduate mathematics classes in the United States is a common practice that seeks to provide consistency in learning opportunities for students while balancing instructional support and pedagogical autonomy. In this study we investigate how the needs of instructors who teach in a coordinated course align with the values and practices of course coordinators. Analysis of 44 survey responses from both coordinators and instructors who teach in a coordinated course identified several coordinator actions and corresponding orientations that align with instructor needs as well as actions that instructors find valuable but that are not frequently occurring. These findings are significant because they can lead to better articulation of the knowledge, skills, and dispositions needed to be an effective course coordinator, which could then lead to the design of professional development aimed at helping coordinators be better able to meet the needs of their co-instructors.
The Department Action Team (DAT) model supports groups of individuals in creating sustainable improvements to education within their department. A DAT, consisting of department members and external facilitators, works collaboratively towards a shared goal. Using innovation configuration (IC) maps we study three DATs, exploring members’ perspectives about their goals, work, and the process of actualizing goals. We examine the degree to which these groups perceived their alignment with the DAT model, and close with a discussion of how IC maps can be used as a tool to assess the alignment of team-based change work to change models.
In this study, we explore notions of HSI servingness and equity in practice as perceived by instructors, facilitators, and students in active learning collaborative, precalculus and calculus problem-solving courses. These courses were part of a grant-funded, scholars program for minoritized student populations. While program outcomes on student achievement, engagement and confidence (discussed elsewhere) are promising, our intent here is to identify and characterize how equity was understood and experienced in practice by those taking and facilitating the courses. Our analysis suggests that instructors and students agree on the learning themes where equity played a role within these active learning courses, but differ in their conceptions of equity within these themes, and in how they saw equity being enacted in practice.
The contributed report highlights the understanding of matrix representations of linear transformations mapping finite-dimensional polynomial vector spaces among university mathematics majors within a dynamic geometry software (DGS)-assisted linear algebra learning environment. Through an analysis of data from in-depth qualitative interviews, framed by the theories of representations in linear algebra (Dorier & Sierpinska, 2001; Hillel, 2000) and the concreteness principle (Harel, 2000), the findings reveal a range of innovative and creative ways in which the students integrated visual and analytical approaches to understand, produce, and interpret matrix representations of linear transformations. The results contribute new insights into the pedagogy of university-level linear algebra, highlighting the growing understanding of the role of visualization in the teaching and learning of the subject. The study also adds to the increasing body of evidence supporting the importance of visualization in linear algebra education.
This scoping review examines the use of student explanation strategies in postsecondary mathematics and statistics education. We analyzed 46 peer-reviewed articles published between 2014 and 2024, categorizing student explanations into three main types: self-explanation, peer explanation and explanation to fictitious others. The review synthesizes the theoretical underpinnings of these strategies, drawing on the retrieval practice hypothesis, generative learning hypothesis, and social presence hypothesis. Our findings indicate that while self- explanation and explaining to fictitious others foster individual cognitive processes enhancing generative thinking, peer explanation have the potential to combine these benefits with collaborative learning. However, explanation to fictitious others have the potential to mitigate some of the negative impacts that may occur in peer explanation, such as more knowledgeable students dominating peer discussions. The efficacy of the methods varies based on implementation, duration, and context. This scoping review contributes to the growing body of literature on generative learning strategies in postsecondary education and provides insights for optimizing the integration of student explanation techniques in within mathematics and statistics.
This study contributes to research on the impact of questions on students’ reading of mathematics and informs design considerations for questions included in workbooks. We used eye-tracking complemented by open-text questions. Participants were engineering and science students from the first to the third year of their studies. We considered proceduralness, conceptualness, and difficulty as dimensions of the workbook questions. We found that what we call a ‘conceptual question’ triggered more attention to the explanation than the example part of the workbook text, but so did what we call the ‘procedural question’. In addition, relative question difficulty may have influenced our results. An implication of our findings is the need to investigate the interrelations of the conceptualness and proceduralness of mathematical questions while also considering the role of questions’ relative difficulty.
Mathematical modeling tasks require modelers to consider both real-world conditions and mathematical properties while constructing models that represent the given real-world phenomenon. While the field is matured with research on the processes involved in constructing models, little research has explored what modelers center their focus on (i.e., notice) while solving modeling problems and how those centers of focus inform the strategies for solving modeling tasks. Through analyzing data from two pairs of undergraduate pre-service teachers’ work on modeling tasks, we found three cases in which PSTs’ centers of focus informed their strategies for model construction. In this report, we illustrate two of those cases and discuss implications for research and instruction on modeling.
This study compares mathematical knowledge for teaching college algebra at community colleges between instructors and math majors without teaching experience, aiming to show that the instrument assesses more than mathematical knowledge. We used an instrument designed to assess knowledge used in doing two tasks of teaching across three function types taught in college algebra, linear, rational, and exponential functions. The Multiple Indicators and Multiple Causes (MIMIC) model was applied to 416 community college mathematics instructors and 85 college students to estimate mean difference in the knowledge between the groups. The higher scores in the MKT-CCA suggests that instructors possess higher MKT than the students. As such, this instrument can be used to identify areas for focused faculty development.
Symbolic forms is a theoretical construct that has been used to study the meanings students have for symbols. One promising application of symbolic forms is to describe how students construct mathematical equations to represent real-world relationships, a skill needed across STEM disciplines. As more scholars use symbolic forms as an analytic tool across multiple contexts, they make adaptions to the theoretical construct by expanding existing lists of symbolic forms. In this paper, we continue this line of work by documenting undergraduate STEM majors’ symbolic forms corresponding to inverse relationships that arose as they modeled dynamic scenarios. We end the discussion with theoretical considerations researchers must make to continue to use symbolic forms to describe students' equation construction.
The mathematical meanings that instructors hold necessarily impact the ways that they teach and their goals for student learning. This study examined five graduate student instructors’ (GSIs’) mathematical meanings related to inverse function with the goal of exploring links between their meanings and the meanings they hoped for their students to construct. I found that the GSIs had multiple meanings for inverse functions that were evoked in different contexts, and that they had trouble drawing connections between these meanings when asked about relationships between them. Similarly, their goals for student learning did not include connections between meanings and were additionally constrained by factors such as the departmental curriculum used and the need to adhere to mathematical convention that students were accustomed to.
Failure and attrition are unfortunately common experiences for developmental mathematics students. These students also frequently experience course repetition and redundancy, though these phenomena are less well-understood. In this mixed-methods study, I aimed to investigate how developmental math students’ attitudes and motivation are impacted by their experiences with relearning and course repetition. Additionally, I aimed to describe how these experiences influence shifts in motivation, resulting in changes to academic behaviors and, thus, performance. Quantitative analyses, though statistically insignificant, warrant future inquiry regarding the attitudinal impact of course repetition. Qualitative analyses reveal how students’ particular aims and learning histories meaningfully mediate shifts in their motivation. The results of this study demonstrate the salience of investigating how repetition impacts students’ motivation, as well as the utility of theories of motivation in the study of developmental math students’ perceptions.
Applying to graduate school in mathematics requires both a desire to attend graduate school, but also an understanding of the application process. The more students know about the process, the more successful they are in their pursuit of admission to a graduate program. We examined mathematics majors’ knowledge of both graduate school and the graduate school application process as part of a larger study examining barriers to students applying to graduate school in mathematics. We also examined the impact of mentors on student interest in graduate school and whether having a mentor has an impact on student knowledge of graduate school. We frame and explain our results using the theory of social capital.
Algebra has been found to be a barrier to college and STEM-major completion, and a contributing factor to unequal access to STEM fields. Conceptual understanding has often been discussed as an important component of mathematics learning; yet more marginalized students often have less access to rich mathematics instruction and to algebra learning opportunities. However, to date no research has investigated the relationship between conceptual understanding and math or STEM outcomes in college. In this research, we explore the predictive validity of the recently validated Algebra Concept Inventory (ACI) to determine whether college students’ ACI scores predict math course grades, STEM-major math course completion, and STEM vs. non-STEM degree attainment as well as whether ACI score explains outcome differences by race/ethnicity/gender.
This study explores Transition-to-Proof (TTP) students’ proof comprehension by analyzing written responses to proof intention and validation tasks. We examined students' use of set-based reasoning as they completed these tasks. Our analysis of 28 students’ written work found that students used set-based tools for logic to support their reasoning. We also found that their explanations in their responses included discussion of the overall structure, individual lines or components, the mathematical veracity of the claim, and a walk through of the proof-text. We concluded that in these tasks students leveraged set-based reasoning for logic to make sense of the context and claim of the proof, supporting them in their overall proof comprehension.
In this study, we analyze interviews with prospective secondary school teachers focused on how an upper division inquiry-oriented content course influenced their beliefs about learning and teaching mathematics. In particular, we examine how students describe their experiences in this course and the extent to which (and why) students see themselves using an inquiry-oriented instructional approach in their future teaching. Using thematic analysis, we found that the ways prospective teachers described their experiences in the course aligned closely with the four pillars of Inquiry-Based Mathematics Education, and the course empowered students to embrace a critical stance to self-reflect on their experiences and to raise their awareness of their agency to change things (for the better) through their own actions and future forms of instruction.
Equivalence relations are a foundational concept in mathematics, yet productive student reasoning about equivalence relations remains largely understudied. Based on the descriptions of productive reasoning about equivalence that we have encountered in the literature, and the results of the data we collected from task-based interviews with advanced mathematics students, we argue that an essential aspect of productive reasoning with equivalence relations is the flexibility to transition between an element-wise, local perspective of the equivalence relation and an equivalence class level, global perspective of the equivalence relation.
This research investigates students' structure sense in algebra, focusing on how they conceptualize and interpret mathematical syntax. Here we focus on how students parse symbolic representations in algebra, in particular, how and why students recognize certain substrings to be unified subexpressions. We use prescriptive conceptions (desired mental images) to guide our noticing of students’ descriptive conceptions (actual mental images) as they engage with conceptual algebra tasks from the Algebra Concept Inventory. The findings offer implications for instruction and curriculum, suggesting approaches that may better support students in developing a deeper understanding of algebraic structures and symbolic reasoning.
In this study we explored the intended outcomes that participants in the SIGMAA on RUME conference had for their conference participation. We conducted in-person, semi-structured interviews at the 2024 RUME Conference and engaging in thematic analysis to arrive at three broad categories of intended outcomes. People intended to achieve specific career-oriented goals such as ‘gaining a line on a CV’ and research-oriented goals such as getting feedback on their work. Finally, participants expressed multiple personal outcomes driven by their sense of the RUME participants as a community that they enjoy being part of. In addition, we found aspects of the conference and community that both support and inhibited the participants in their pursuit of their intended outcomes. The analog to the perception of a strong community is that participants find it difficult to become part of the intellectual and social conversations.
As computation becomes increasingly central to mathematics education, instructors must balance competing forces when choosing which computational tools to use in their courses. This is compounded in probability and statistics where computation is widely used. Grounded in a social constructivist perspective, we believe that tools mediate our activities and that different tools play different mediational roles. As such, this study explores how different computational tools mediate undergraduate students' mathematical activity of argumentation. Using Toulmin’s argument model, this research investigates how two classes in probability and statistics using different computational tools, R or Minitab, performed on a mirrored assignment. Through analysis of students’ assignments, a difference emerged across the classes use of visuals. Our findings suggest Minitab promoted more deliberate consideration and use of visuals than R, leading to a difference in arguments produced by the students.
Mathematics tutoring centers are places where students learn math, but little work exists about learning from tutoring. Much of the research about tutoring is from an observer’s perspective. We build on this work by presenting a study of tutors’ experiences and actions when tutoring students with written and online homework, as described by tutors. We also suggest ways the Instructional Triangle could be adapted to account for the presence of a tutor as an agent in learning.
The Carnegie Classification sorts institutions by their different styles of education. Two prominent types of institutions are high research activity universities (R1) and liberal arts focused colleges (LA). Institutional characteristics may positively (e.g. Eide et al., 1998) or negatively (e.g. Astin, 1997) affect graduate school aspirations. We analyzed responses from a national undergraduate mathematics major sample using chi-squared and Mann-Whitney U tests to identify differences between students’ knowledge about graduate school and its application process by these two types of institutions. Using this same sample, we used chi-squared tests to explore the differences between departmental (professors, advisors, and mentors) support by institution type. We interpret these results with the theories of social and cultural capital and offer suggestions for future research investigations.
This paper presents the results of a study where a student’s concept image and concept definition of periodic functions were explored. The findings suggest that a student thinks about periodic functions as visual repetition of patterns within a function’s graph. He also analyzes output values, particularly at distinguishable points such as maxima, to determine whether a function is periodic. His concept image and concept definition include the idea that a function’s period represents the length between any two points where the output values are the same and the pattern repeats throughout the domain.
This study explores how linear algebra students reason with three entities: magnitude, unit size, and coordinates in change of coordinate systems during teaching experiments. Two conceptual frameworks are employed in this study. The first framework, the coordinate system framework, was developed prior to the study and is used to frame a larger study that includes clinical interviews and teaching experiments. The second framework, the entities framework, was developed from the results of clinical interviews and is used to analyze student work from the teaching experiments. Students interacted with a GeoGebra applet that creates coordinate planes using matrices. The results present how students establish relationships among the three entities while working with the applet and how they represent their reasoning through matrix entries.
University teacher preparation programs extend significant efforts to support the development of secondary teachers’ competence in teaching mathematics. However, it remains an open question whether or to what extent beginning mathematics teachers enact ambitious practices in their teaching – specifically, engaging students with reasoning and proving. We report on one teacher, Nancy, who, after two years of autonomous teaching seized the opportunity to design and enact a “proof unit,” by utilizing materials from her undergraduate studies. Using the framework of practical rationality and four professional obligations, we examine Nancy’s discourse over various time points on her professional journey, focusing on how she held on to the ideas from her university training, culminating in the reasoning-and-proof unit. Implications for strengthening mathematics teacher preparation beyond university programs are discussed.
Students' beliefs influence the ways in which they learn mathematics. These beliefs can be categorized in different ways and related to growth or fixed mindsets towards mathematics. In our work, we seek to better understand connections between students’ beliefs and mindsets about learning mathematics. We present case studies of two undergraduate students who were enrolled in a first-year college algebra course who demonstrated a shift in mathematical mindset. Our findings highlight the complexity of students’ beliefs and how students can hold both growth and fixed mindsets towards mathematics. We explore how mindsets can be tied to specific categories of beliefs and how those beliefs can influence a student’s overall perspective about learning mathematics. We conclude that mindset interventions and future research should consider how to leverage these mixed mindsets to help students foster more growth-oriented beliefs about mathematics.
We report on the initial findings from a project aimed at enhancing equitable group work in undergraduate proof courses through the design and implementation of group-worthy tasks. This mixed methods study reports on three group-worthy tasks implemented in a topology course. We investigated the extent to which student interactions in these small groups reflect participatory and relational equity in relation to perceived academic status. Quantitative results indicate patterns in which the group-worthy tasks may disrupt students with high status from dominating talk turns taken, the amount of talk turns taken, and number of words spoken by the students, or neither depending on the tasks. Qualitative results explored relational inequity via silencing, ignoring, or no talk during these three tasks. Our initial findings suggest that group-worthy tasks alone may not guarantee equitable participation or relations in group interactions.
Research on combinatorics education indicates a crucial need in mathematics education to address students’ ways of thinking at a level that benefits a deeper understanding of how students conceptualize counting problems. Attending to the ways in which students approach and interpret counting problems reveals the subjectivity in their decision makings. In this study, I report data from written works submitted by thirty five prospective high school teachers as they considered a disagreement between two possible responses to a combinatorial problem. Defining subjective combinatorics as a subjective aspect of students’ interpretations of sets of outcomes facilitates describing students’ counting activity. I propose subjective combinatorics as a construct to shed light on our understanding of students’ counting activities, focusing on the subjectivity inherent in their combinatorial thinking.
The increased use of educational technology has raised questions about technology equity in mathematics education research and pedagogy. This paper provides a subset of findings from a larger systematic review, which aims to investigate literature on equity and personal device use in undergraduate mathematics courses from 2009-2023. For the systematic review, literature was searched and screened for information on personal device access, utilization, and outcomes. This paper reports on the subset of findings specific to the utilization of video resources in mathematics classes. Through the synthesis process variable information regarding student outcomes and student perceptions of use was pulled from 82 studies. Findings suggest that the use of video resources correlates to either higher academic outcomes or no difference in academic outcomes. Additionally, student perceptions suggest that video resources can afford an equitable educational experience in a variety of ways, but elements of distraction and communication can serve as limitations.
Many undergraduate mathematics departments are seeking to improve equity in their programs; however, they may struggle to translate these goals for equity into action. In this study, we used Cultural Historical Activity Theory (CHAT) theory to investigate the evolution of a Networked Improvement Community (NIC) focused on improving equity within their mathematics department, highlighting ways that NIC discourse evolved to support more critical, action- oriented activity. By examining field notes of meetings, documentation (e.g., agendas), interviews, and journal entries from NIC members, we identified key components of the NIC activity system. Our analysis revealed ways that NIC leaders mediated tensions between their object (goals) and implicit discourse norms governing NIC conversations. Specifically, we found that to make progress toward their goals, NIC leaders needed to establish an explicit rule to take certain topics “off the table” in NIC meetings. This rule facilitated discourse that was more action-focused than previously observed.
In this paper, we explore the role that silence plays in the experiences of multilingual students of color within undergraduate mathematics classes. The poetic narratives presented in this paper focus on exploring the role of silence in students’ experiences. While silence often reflects marginalization (e.g., being silenced), it can also function as a mechanism for navigating different spaces, and as a form of resistance. As such, we take an asset-based and anti-deficit framing to explore how silence interacts with students’ cultural capitals, including, for example, navigation capital, resistance capital, aspirational capital, and language capital. We use poetic transcription to present students’ stories using only their own words.
The implications for students and teachers of educational digital tool design have thus far been under-examined. We find it important to problematize and bring attention to relationships between the explicit or hidden choices of digital tool designers and implications for the possible learning taking place with the tool. Exploring one aspect of these relationships, we examined extant digital tools within the undergraduate mathematics curricula. Grounding our analysis in Piaget’s genetic epistemology, we took on the perspective of the epistemic user and identified features of extant digital tools that hold implications for the learner-user experience. With this we generate initial categories that contribute to a user-oriented phase of a framework for digital tool design and discuss implications for further research.
This study investigates how instruction in the base-8 number system in a Calculus Course for pre-service elementary teachers (PSTs), affects PSTs’ conceptual understanding of place value. Despite the importance of number sense and numerical operations in the K-12 curriculum, research shows that many PSTs struggle to grasp place value conceptually, often focusing on procedural rather than conceptual knowledge. This study uses a convergent mixed-methodology approach with quantitative pre- and post-tests and qualitative clinical interviews to analyze PSTs' understanding of place value after instruction in base-8. Findings suggest that instruction in base-8 improves the sophistication of PSTs' place value conceptions, though verbalizing that understanding remains a challenge for some. These results highlight the need for a focus on both conceptual understanding and consistent language use in undergraduate teacher preparation programs.
Interest has been growing in how students learn concepts in complex analysis (Troup, 2015; Hancock, 2019). A concept generalized in complex analysis is exponents and logarithms, yet little is written on how students leverage their existing knowledge to the complex case. In this paper, I present data from task-based interviews via the lens of APOS theory to present two students who leveraged their knowledge of real-valued exponential and logarithmic operations to varying degrees. I present data from a single question asking them to compare the magnitude of complex numbers and argue that their schemas for the two subjects were separate.
Following the work by Shultz et al. (2023), I present data related to surveys and interviews of math education researchers from underrepresented cultures in the wider literature. I present findings related to constructs they deem important in math education and have not seen widely represented in current research.
Mathematics education scholars have asserted that proof can be viewed as a unique genre of communication with specific norms and conventions. Further, some have argued that these norms and conventions are gendered and may dissuade women from participating in advanced mathematics. However, these assertions for the most part have not been investigated empirically. In this proposal, we systematically analyze the linguistic features of 90 proofs from 15 textbooks in advanced undergraduate mathematics. Key findings include that many feminine ways of communication rarely occur in proofs, including hedges, references to emotions, and intensive adverbs. Some masculine ways of communication, including the use of directives, are common in undergraduate proofs. Implications of this research are discussed.
Students’ identities, orientations, and experiences in mathematics courses shape their participation and longevity in the discipline. Yet, undergraduate mathematics instructors do not often integrate pedagogy or curricula that help them build knowledge of their students’ math identities. We believe that in many math courses this is a problem of time and scale. We propose that language processing methods can help alleviate this tension by giving instructors a baseline understanding of their students’ math identities. In this study, we applied sentiment analysis to 28 mathematics “origin stories” written by undergraduate pre-calculus students. We compared the predictions from four language models with qualitative codes applied by two mathematics education researchers. We conclude this paper by discussing the strengths and limitations of sentiment analysis in our data, and discuss implications for undergraduate mathematics researchers and practitioners who may wish to apply these methods to their work.
Too often college mathematics classes become a gatekeeper that pushes students off of STEM pathways. This study looks at an example of a community college mathematics instructor who appears to be successful in supporting students to stay on a STEM pathway through creation of a warm, caring classroom environment. Using the lenses of a caring relation and a teacher as a ‘warm demander’ this case study examines what pedagogical practices he used to create and sustain this environment. We illustrate in the findings how the instructor used his emotional intelligence and skill in relating to students to sustain a caring, warm classroom. He also incorporates metamessages and ‘off-topic’ talk in his classroom which demonstrate care to students. This case study serves as a beginning point in thinking about the social and emotional work of teaching and we end by proposing a model for caring instruction in community college mathematics classrooms.
We created and piloted a playful calculus-based activity called Major Tom with preservice secondary mathematics teachers over two classroom sessions. Players create sequences of jetpack boosts (accelerations), controlling an astronaut’s movements to arrive at a pick-up location at a particular time. The learning goal was for students to draw upon and enhance their knowledge of integration to determine velocities and displacements. We found that the imprecision of the system’s feedback inhibited the students’ productive use of their formal calculus knowledge, while on the other hand an instructor-led structured discussion about velocity graphs supported it. Neither of these features of the playful math environment much affected the students’ playfulness, but a third feature, the opportunity to create and solve your own challenges, fostered significantly increased play and laughter. This contributes to our broader goal of how to elicit play with undergraduates while simultaneously engaging and building their formal mathematics knowledge.
We examine undergraduate calculus students’ reasoning about the continuity of a function at a point. The study was conducted in the context of an NSF-funded project that supported the redesign of Calculus I recitations at a large public university. One aspect of the reform effort involved introducing conceptually rich, collaborative activities into Calculus recitations. One such activity focused on the concept of continuity at a point, with one particular task probing students’ understanding of the definition of continuity at a point using true-false statements. Data from 84 student in-class worksheets were analyzed with a focus on the type of reasoning students used to respond to these questions. The results reveal a variety of graphical, logical, example-based, and rules-based strategies.
The concept of equivalence is fundamental to problem-solving in mathematics. Particularly, an oft-utilized feature of equivalence is substitution, in which the mathematics doer replaces one object with an equivalent object. Equivalence in general, and substitution in particular, have been identified as common threads that can tie together different areas of mathematics, but do students notice these connections in their own mathematics? In this paper, I report on task-based clinical interviews with students exploring the commonalities they identify among tasks implicitly involving equivalence relations. In particular, I discuss themes that students explicitly identified as relevant when answering the interview tasks: (1) substitution equivalence and (2) a distinction between students using a known equivalence for substitution or introducing their own equivalence.
This study explores the changes in a graduate student teaching assistant (GTA)’s teaching actions over two semesters in an active learning Calculus I course. Cole, a GTA with little teaching experience and no prior experience with active learning, was paired with an experienced undergraduate learning assistant (LA) in his second semester teaching in a purposefully designed active learning-based Calculus I course. Cole’s teaching actions varied more when paired with the experienced LA, including short periods of lecture and classroom discussion with the facilitation of group work. This case study provides insight into how an experienced undergraduate LA can be an agent of change in an active learning-oriented classroom.
Graduate student instructors (GSIs) in mathematics play a pivotal role in shaping undergraduate education and are the future of collegiate mathematics faculty. As part of their development, GSIs are expected to engage in teaching-focused professional development (TPD), particularly in evidence-based strategies like Active Learning (AL) methods. However, higher education is only beginning to explore how to effectively measure GSIs' growth in teaching skills through such TPD. This study examines the learning process of 47 novice GSIs from three universities, specifically focusing on their evolving understanding of AL before and after participating in TPD. By analyzing the GSIs' own definitions of AL, the research highlights changes in their knowledge and alignment with the intended TPD outcomes. The findings provide insight into the effectiveness of TPD on AL, while also offering recommendations for structuring future evaluations of TPD impact on GSI teaching knowledge and skills.
Graduate students’ values related to teaching and learning are critical in shaping their professional identities and future careers as faculty members. These values encapsulate instructors’ convictions about what is important or worthwhile as it relates to mathematics teaching and learning. Research suggests that mathematical discourses reflect and empower dominant masculine values. Thus, we explore three women-identifying graduate students’ values related to teaching and learning mathematics to highlight their voices and emphasize ways in which their values align with or challenge dominant masculine values. Utilizing interview data, we found four values across the three participants: collaboration, student-instructor perceptions, humanizing mathematics, and compassion and care for students’ wellbeing. We discuss how these values interact with dominant masculine mathematical discourses and the ways in which these values may help foster equitable mathematics pedagogy.
Projects to improve teaching as a means to improve learning need good tools to measure shifts in the use of effective research-based instructional strategies. We describe a survey tool to assess active and collaborative learning practices in college STEM by asking students to report what practices are used commonly in their course. Student reports correlate with their experience of the learning environment, and with their self-reported learning outcomes. Student means by section are corroborated by instructor-reported use of these practices and external observations.
In modeling physical phenomena and solving associated quantitative problems, students reason about constants in various ways. These symbols represent elements of mathematical structures with embedded algebraic operations and are imbued with contextual physical meaning. In this paper, we report on what meanings for the constant ħ/2 students conveyed in a quantum mechanics expectation value problem from a spin-½ context. Through analysis of interview data with 12 students from two different universities, our analysis documents eight different meanings for ħ/2: part of expression in problem setup, part of calculation, result (either consistent or not consistent with unit interpretation), eigenvalue, measurement, average, and bound. This analysis highlights both the cognitive complexity of reasoning with symbols and the impressive ability of students to flexibly and fluidly leverage multiple meanings for symbols at this content level.
This study explores the crucial issue of attrition in the undergraduate mathematics major. In this report, we thoroughly examine open-response survey data from seven former mathematics majors describing why they chose to major in mathematics as well as when and why they chose to leave. We situate these data within findings from the 2019 study Talking About Leaving Revisited. Findings from the present study indicate students may leave mathematics because they are intrinsically drawn to a new discipline, or they may feel pushed out of mathematics due to perceived poor instruction and/or lack of outreach and support.
Addressing equity and opportunity gaps in undergraduate education requires change across systems, including the pedagogies and orientations of the undergraduate faculty whose courses and departments perpetuate these gaps. Hence research on how undergraduate faculty notice quantitative and qualitative evidence of inequitable experiences has the potential to advance reform efforts. We conducted an interview-based study of 5 mathematicians who were presented with data analytics indicating equity gaps in proof-based courses at their university, and statements from minoritized students in these courses about their experiences. We found that the faculty may hold split inclusive and exclusionary frames about students and mathematics; that there may be a persistence of personal experience as an interpretive lens; and that qualitative and quantitative evidence may elicit different frames of instruction and students. We discuss these findings through the lens of a noticing framework.
In this article, we share results from a series of interviews with a pair of undergraduates using an applet designed to support students’ understanding of determinants. We identify four generalization clusters that the participants developed, and we share our analysis of the activity types that led to these generalizations. We used Ellis et al.’s (2022) Relating, Forming, and Extending framework to characterize the participants’ generalizing activity. We share results from this work and discuss the implications for applet design, instruction, and future research.
We explore undergraduate students’ perceptions of and the impact of implementing active learning in calculus recitations. Analysis of Likert-scale items indicate that speaking up in recitation helps clarify students thinking, suggesting that calculus instructors should create opportunities for students to speak up in the classroom. Qualitative analysis of two open- response questions indicates an increasing trend over time of students responding that working on problems in small groups was one of the most helpful aspects of recitation. Although students responded that dissatisfaction with group work was one of the least helpful aspects of recitation, this response was never as prominent as group work being the most helpful aspect. An independent two-sample t-test on students’ course grades indicate no statistically significant difference (p = 0.14) between the treatment group (active learning) and control group. Results suggest that active learning with structured group discussions enhances student engagement and collaboration in learning Calculus concepts.
Math placement policies are intended to match incoming freshmen with an appropriate math course. At our four-year institution, the same placement test is used to place students into nine courses across three pathways. In this paper we use institutional data to analyze the effectiveness of test-based placement into different courses requiring similar (and sometimes the same) cutscore. We use the classification power metric to evaluate and compare the effectiveness of three placement policies. We report that that placement tests provide minimal information about students’ readiness for mathematics courses that are not algebra intensive, such as basic statistics.
We report results from a longitudinal case study of Elementary and Intermediate Algebra students’ perceptions of what it means to “relearn” algebra, and the consequences of such perceptions on relevant behaviors and affective states. This data comes from a larger project investigating student experiences relearning individual topics and reflections on relearning across two semesters of developmental math courses. We found that students often conflated their perceptions of algebra as a familiar and “simplistic” subject with the notion that they understood it, leading to the idea that relearning it should be easy. This perception was held even in the face of poor course performance and was used to motivate unproductive patterns of behavior. In identifying the unique factors that informed and reinforced students’ perceptions of relearning, this report provides useful insights for educators who are currently grappling with high rates of failure and attrition in such courses nationwide.
In this report, I present how eight experienced college calculus instructors used exploring and conjecturing to frame their instructional tasks for introducing derivatives. During semi- structured interviews with each instructor, I prompted them to propose instructional tasks for introducing derivatives. The tasks were broken down into their smallest calculus-specific problems, which I refer to as calculus-specific task units or CTUs. Using the notion of instructional situations by Herbst (2006), I categorized the exploring/conjecturing CTUs into a single instructional situation: Exploring or conjecturing to verify statements or relationships. I further describe how the CTUs were invoked in the tasks and the kinds of mathematical works they were expected to engage students in. To conclude, I discuss why the exploring/conjecturing CTUs revealed in the study either did not, or could not, converge to proving situations.
Linear algebra is an important topic with many in-demand applications. In this paper, we consider a set of digital interactive figures created for introductory linear algebra students. The figures were designed to scaffold student experimentation with linear algebraic objects in the process of making observations and conjectures. We analyzed a collection of student reflections written about their experience and observations from the interactive figures through the theory of instrumental genesis. Based on the results of this pilot study, we believe that these interactive figures can be effective in promoting cycles of observation and conjecture as pre-proof activity.
As students advance in undergraduate mathematics courses, the formal (e.g., textbook definitions and proofs) and informal (e.g., metaphorical) language they encounter becomes increasingly loaded. Additional work is needed to unpack how learners can engage in semiosis “meaning making” processes beyond surface marks, symbols, and verbiage. In this paper, we conduct semiotic bundle analyses to explore the potential of a full range of embodiments in meaning making. Our results show how embodiments can be leveraged in a more advanced abstract algebra setting to mediate semiotic conflicts between students’ and institutional/communal signs and expand meaning-structures. We present more specific meaning tuning and elaboration strategies that utilized embodiments and were identified during our analyses.
In the last 20 years, mathematics education research has taken a more critical lens (Gutiérrez, 2013), in which researchers focus on salient features of students’ identities (e.g., race, ethnicity, gender) to analyze the structures within mathematics and schools to motivate change to power structures and systems that oppress such groups. In recent years, there has been a notable change in the types of submissions to the RUME community. This paper details a meta-analysis of the RUME proceedings from 2019 to 2024. Using 60 keywords related to racial identity, we sorted papers based on how they discussed race and its centrality in their work. A total of 33 papers were identified as directly engaging work that focuses on race. We draw conclusions of what may be missing in current research, and the ways we may more meaningfully engage in race-conscious research in mathematics education.
Although some contemporary definitions of inquiry learning in college mathematics include instructor attention to equity as a core precept, evidence suggests the practice of inquiry-based instruction in proof-based undergraduate mathematics courses entails many decisions whose implications for inquiry and implications for equity and access may not be mutually reinforcing. We present the results of our analysis of over 40 design and facilitation decisions made by the second author as he taught an undergraduate-level number theory course for high school students in a summer residential mathematics program. In our analysis we identify some categories of instructional decisions that have both inquiry and equity implications. We also discuss in detail a scenario in which the instructor was able to choose a course of action that simultaneously supported inquiry and equity goals of the class, and one in which the goals of inquiry and equity appeared to be in tension.
In many undergraduate math courses, students are expected to interpret and use conditional statements without prior training in logical inference. In this paper we report on a portion of the third phase of a project exploring students' interpretations of a conditional statement in Probability relating covariance and independence. Specifically, we discuss the coding and analyzing of student written work in class and on assessments. We explain how this work was influenced by prior phases of the study. Quantitative results show no statistical difference between responses of students with and without prior training in logic. We do, however, see differences in students’ responses overall as the semester progresses.
Fewer women than men choose to major in STEM, and women leave STEM majors at a higher rate than men – especially after taking Calculus. Prior research identifies low sense of belonging (i.e., feeling like an accepted member of an academic community) as a key reason why women decide to leave STEM majors. Scholars have identified factors that contribute to one’s sense of belonging, including students’ perceived competence and social connectedness. This study explores ways in which engaging in active learning opportunities might support women’s perceived competence and social connectedness, and in turn, their sense of belonging. Women’s survey responses suggest that active learning supported their sense of belonging indirectly via their perceived competence and social connectedness, as well as directly.
This mixed-methods study evaluates the mathematical autobiographies (MABs) of STEM students to evaluate their relationship with mathematics and influential actors in their math life stories. 135 MABs were collected from students enrolled in Calculus 1, Abstract Algebra, and Topology, taught by the same instructor. Analysis revealed that students mostly had consistently positive experiences with mathematics (PME). Hypothesis testing showed that students enrolled in upper division courses and math majors had a higher likelihood of describing PMEs and that Asian students described PMEs at a lesser rate than their peers. Instructors and family members were most frequently named as influential actors, with K-12 instructors appearing more often than college instructors. These results indicate that STEM students have had primarily positive mathematical experiences, which aligns with their desire to pursue STEM degrees.
Calculus with Applications in the Life Sciences (Biocalculus) is an impactful and required course for all STEM majors at our institution, a large public university. A disproportionately high percentage of Latinx and students from low-income households (EOP) earn failing grades in this course. This study examines student learning outcomes and experiences based on project assessment (N=152) and student survey (N=130) to answer the following questions: (a) Did changing a course structure based on working within a teaching team change students’ agency? (b) Did minoritized students achieve similar levels of conceptual understanding as their counterparts in the course, when the curriculum was designed with project-based learning situated within the teaching team? We found that the inclusive course structure provided minoritized students with a supportive environment and positively impacted their learning. We believe our specific implementation can be helpful to many large public universities with diverse student populations and relatively limited resources.
The operation of subtraction, though emphasized at the elementary level, has a central role in expressing key ideas in Calculus. In graphical representations of Calculus concepts, a magnitude interpretation of a difference, that b–a gives the distance between a and b, often supports students in making sense of the representation. A magnitude interpretation uses a determine the difference model of subtraction, in which subtraction find how much a differs from b. Using exploratory teaching interviews, we investigated the models of subtraction that nine undergraduate students used on tasks designed to support them in using a magnitude interpretation. We found four distinct models of subtraction that students employed in these tasks. We describe these models and associated observable behavior. We discuss implications of these findings for the teaching and learning of Calculus and future research directions.
The mathematics education community has examined how students conceptualize repeating decimals. Yet, there is potential for further investigation of how undergraduate students conceptualize the addition of repeating decimals. This paper explores pre-service teachers’ conceptualization of repeating decimals. Each participant answered three open-ended questions, and our data consisted of these written responses. Our analysis highlighted that the school conventions hindered many participants from seeing how to add repeating decimals without truncating them. Initially, most participants viewed repeating decimals as processes that approximate rational numbers. Many participants came close to considering repeating decimals as mental objects rather than processes. Our data indicates that asking participants how they would check if they had correctly added repeating decimals enabled them to overcome epistemological obstacles that hindered them from seeing repeating decimals as numbers.
This study investigates the academic journeys of first-generation, low socioeconomic status (SES) women in mathematics, focusing on their sense of belonging, perceptions of competence, and career aspirations. Using narrative inquiry, the research draws on interviews and reflections to illuminate how isolation, identification as “rule followers,” and a lack of mentorship shape their experiences. The study adopts Wenger’s (1998) mode of belonging framework—engagement, alignment, and imagination—to analyze how these students interact with their mathematics community and envision their roles within it. Findings highlight the urgent need for mentorship and career guidance tailored to underrepresented students in STEM. This research provides insight into institutional strategies for supporting the success of first- generation, low SES women in mathematics.
Despite reforms, some College Algebra students are still “at-success”: they can be identified early in the semester as having a higher likelihood of not passing or completing the course (Wakefield et al., 2018). This interview study of 3 at-success College Algebra students explores the hypothesis that there are pedagogical and environmental factors to whether at-success students engage in help-seeking behaviors that could increase their chance at success. I examine: What affordances and barriers to help-seeking do at-success students describe when talking about performance in a college algebra class? Karabenick and Berger’s (2013) process of help-seeking was used as a framework for the themes found using interpretative phenomenological analysis. I propose an adaptation and extension of Karabenick and Berger’s framework as a result of this study.
This study investigates student experiences in undergraduate mathematics education, highlighting the importance of addressing inequities related to race and gender in student success rates. Focusing on support courses for Precalculus and Calculus I, where students participate in regular Supplemental Instruction sessions and an online course led by the university Math Equity Coordinator, I examine how these interventions influence student success beyond traditional measures such as grades and persistence to include the impact on students' confidence, sense of belonging and the perception of instructional practices. Results from this mixed-methods analysis show that students enrolled in the support course report higher levels of confidence and a stronger sense of belonging in mathematics. Interviews further emphasize the critical role of instructional practices have in fostering these positive outcomes. These findings suggest that to investigate the success of at-risk students effectively, researchers ought to broaden the scope of success indicators to include identity-related aspects of student experience.
In this paper we analyze the reasoning of five students who were playing a 3-dimensional digital video game called Vector Unknown: Echelon Seas. The intention of Stage 4 of the video game is to give students experience with Numeric and Geometric representations of linear combinations of vectors in the context of a pirate throwing a grappling hook to reach an anchor. We illustrate three types of Numeric reasoning, and four types of Geometric reasoning students used to solve these puzzles. For each reasoning example, we discuss issues related to which vectors the player should choose so that the anchor is within the span of the vectors, and which scalars the player should choose so that the grappling hook ends precisely at the anchor.
Research on teaching and learning proofs in linear algebra is scarce. This paper used the instrumentation genesis approach to examine students’ proofs and MATLAB codes. The students were studying a proof-based second course in linear algebra. The instructor included application-focused labs and a final project as part of this study. One of the labs required a proof validation using MATLAB. This lab, a proof question from the midterm exam, and an exit survey were analyzed in this paper.
Lists are a foundational way to reason about and solve counting problems, yet it is unclear to what extent their role should continue for advanced learners and what that role should be. I draw on case study data of two undergraduate students who used lists repeatedly as they worked through a sequence of combinatorial tasks. I demonstrate that the students’ use of lists refined over time, and their representations developed from outcome-oriented to process-oriented, thus allowing them to draw on their knowledge of lists for increasingly sophisticated problems. Further, the forms of their solutions were impacted by their use of lists. I argue that these data shed light on broader questions of the role of representations in solutions to combinatorics problems as students develop combinatorial facility.
This study aimed to gain insight into students’ experiences of engagement in undergraduate Precalculus. A series of interviews were conducted with a group of 12 college students to solicit their lived experiences of engagement, using methods of hermeneutic phenomenology. The analysis uncovers that some of the key factors influencing engagement include the physical layout of the classroom, the nature of student-instructor and student-peer relationships, students' comfort level with the course material, and their interpretation of grades as formative feedback. Insights gained from these findings can provide us with important information about how students engage in college Precalculus and suggest ways to improve the teaching of the course.
A key objective of academic mathematics courses in teacher education is to enhance teachers’ awareness of disciplinary values and principles, ensuring alignment between mathematics education and the discipline. However, research indicates that this goal is often not met. A possible explanation is that mathematicians and teachers have different perspectives regarding the discipline’s role in mathematics education. This study examines these perspectives in three extreme cases, where mathematicians and secondary mathematics teachers disagreed on the legitimacy of secondary mathematics activities from a disciplinary standpoint. Analysis of professional obligations that underlie the mathematicians’ and teachers’ reasoning reveals distinct interpretations of the obligation to the discipline and of its implications and significance in consideration of other obligations. These findings emphasize that teacher education should go beyond fostering awareness of disciplinary values and principles, and support teachers in learning how they can apply this understanding effectively in school to guide and inform their teaching.
Collaboration within mathematics has been established as being effective in providing students with crucial opportunities to develop critical thinking, effective communication, and teamwork skills. By engaging in group problem-solving and shared learning experiences, students may gain deeper insights into mathematical concepts and learn to approach challenges from multiple perspectives. However, there remains a need for a reliable instrument to capture students' preferences for collaboration. This study aims to develop and validate the Collaborative Preferences for Learning Mathematics (CPLM) scale to measure student preferences for collaboration. Exploratory factor analysis revealed a single-factor structure, and a confirmatory factor analysis conducted with a separate sample (N = 243) demonstrated a good model fit. Further testing established the scale’s strong invariance, confirming its ability to reliably measure collaborative preferences over time. The CPLM offers a valid and reliable way to capture student preferences regarding collaborative learning in mathematics.
Difficulties with fractions are well-documented for undergraduate developmental mathematics students. As such, fractions are a known gatekeeping topic for this demographic. However, research on fraction understandings for this population is scarce. In this paper, I synthesize relevant literature regarding undergraduate developmental fraction understandings and related K–12 fraction literature. I then report findings from task-based clinical interviews to share examples of various go-to strategies these participants utilized as they worked through fraction tasks. I close with a discussion relating these findings to extant literature and propose future research directions.
The research reported in this paper examines undergraduate students’ mathematical thinking while engaged with an intellectual need-provoking (IN-P) task. Our central goal is to infer design characteristics and strategies for the pedagogical implementation of IN-P tasks that orient students to confront the key idea that the task was designed to necessitate. The results of our analysis demonstrate that provoking a cognitive state of intellectual need is nuanced and complex and extends beyond thoughtful task design.
As part of a larger participatory, arts-based study, six STEM students engaged in sustained arts creation and reflection across their curricular experiences as a form of holistic, multimodal, and multisensory aesthetic critique of their cross-curricular mathematical experiences. In this contributed report, I detail instances where students’ perspectives on curricular coherence (i.e., their aesthetics of coherence) deviated from privileged forms of disciplinary (logico-rational) coherence in mathematics education. In doing so, I highlight the exclusionary politics of aesthetics that result from a narrow adherence to one perspective of curricular coherence and the epistemic harm this can inflict on students in relation to how they position themselves as learners and doers of mathematics. These examples suggest a need for additional criticality concerning our often-unquestioned definitions of curricular coherence and acknowledgement that mathematical coherence seeking is governed by both aesthetic and logical forces.
Much of the work in covariation, to date, has focused on “large-scale” themes, such as broad reasoning levels and behaviors. We propose that covariation research could be greatly enhanced by attending to the “fine-grained” cognition happening in students’ covariational reasoning. We propose that covariation can be considered as a coordination class, which models knowledge as a complex cognitive structure wherein an emergent system of knowledge resources are activated in response to context and cueing. We explain what this perspective on covariation would look like, and we provide an illustrative vignette to explore the power of this perspective.
This work provides a theoretical grounding for proof education studies that utilize proof comparison tasks. We leverage the notion of intertextuality, borrowed from genre theories, to discuss mathematics education theory and methods. Intertextuality involves the implicit or explicit comparison of two or more texts with regard to a given genre. We discuss how this comparison can serve explorations of (student) conceptions of the genre of proof and argue that intertextual comparisons lead to novel insights into (student) conceptions of the genre. Specifically, we illustrate how someone's comparison of two (or more) justifications relative to the genre of proof can provide insights into how they view (a) proof as a genre, (b) which attributes of a justification are relevant to that justification's (potential) status as a (good) proof, (c) the relative importance of those attributes, and (d) the reasons for that relative importance.
Over the past decade, research on students’ understanding of integrals has grown significantly. While previous studies have noted students’ difficulties in coordinating the “product layer” with definite integrals in applied contexts, they often overlook the diverse knowledge resources students use in their moment-to-moment reasoning. Additionally, effective methods for analyzing this complexity have been lacking. This paper introduces new methods for examining students’ reasoning about the “product layer”, !(#) ∙ &#, and definite integrals, combining a modified dynamic conceptual blending diagram and an expanded Visualization/Analysis/Physics (VAP)- model. We demonstrate these methods with an example of an integration task, illustrating their application and impact on integration research (see Sus & Izsák, 2024b for further examples). This contribution advances the literature on integration research by providing a more nuanced analysis of students’ reasoning.
Although much research has made the case for the value of students’ making sense of others’ solutions, explanatory mechanisms for how such learning occurs are lacking. In this theoretical report, we consider how students’ making sense of others’ mathematical solutions may support learning from a radical constructivist perspective. We elaborate on radical constructivist constructs–social goals, cognitive perturbations, and reflective abstraction–and use these constructs to model how engagement with others’ mathematical solutions may engender learning. We illustrate our model with a task we designed to promote students’ meanings for spatial coordinate systems. We conclude with implications for research and teaching.
Social Cognitive Career Theory (SCCT) has been extensively employed to elucidate the enduring gender differences in mathematics-intensive fields, with a particular emphasis on the complex interplay of motivational factors and extra-personal influences contributing to the underrepresentation of women. Although a plethora of empirical studies corroborate SCCT, three crucial aspects for refinement have come to the fore. First, the theory should place a more substantial emphasis on how cultural and contextual diversity influences academic choices. Second, given the dynamic nature of motivation, which evolves over time, more longitudinal analyses are imperative to capture their temporal trajectory, in contrast to the predominantly cross-sectional empirical studies. Finally, considering the intricate interplay between emotion and motivation, integrating the dimension of emotion into SCCT would significantly augment its explanatory power and provide a more comprehensive understanding of academic selection processes.
Researchers have argued that interpreting definite integrals and their notation as summations of small amounts of a target quantity (Adding Up Pieces) is especially productive for applied problems. Jones and Fonbuena (2024) extended research on Adding Up Pieces by providing an analysis of integration by substitution grounded in quantitative reasoning. The present theoretical paper develops a complementary justification that makes three contributions. First, whereas past research on integration has rarely foregrounded explicit, quantitative meanings for multiplication, the analysis presented here is based on connections between multiplication and coordinated measurement with two units. Second, the analysis presented here uses coordinated measurement to explain how, across diverse situations, small amounts of a target quantity can be understood as products. This supports reasoning about Riemann sums and areas as approximations for definite integrals. Finally, the analysis provides a new justification for integration by substitution that relies on rectangular areas and inversely proportional relationships.
This study proposes the critical perspective on AsianCrit (Asian [American] Critical Race Theory) and suggests a new framework for potential research and practices in mathematics/STEM education regarding Asian identities. This study reviews and critiques the traditional views on AsianCrit for postsecondary mathematics/STEM education. Particularly considering Asians’ diverse immigrant and citizenship status in STEM, gender identity, ethnicity, and more social identities, this study addresses the complicated relationships between Asian stereotypes, the Model Minority Myth, the intersectional identities of Asians, and the possible ways of empowerment.
Bandura’s self-efficacy theory notes that self-efficacy determines how much time and effort students will put toward their work, meaning higher self-efficacy will lead to students working harder on their coursework. It should thus be in every mathematics instructor’s best interest to understand how they can positively impact a student’s self-efficacy related to mathematics. This theoretical report examines the intersection between an actor-oriented perspective of transfer of learning and Bandura’s self-efficacy theory. I then propose a merging of the two through the concept of actor-oriented self-efficacy, which emphasizes using actor-oriented transfer methods to better understand students’ mathematics self-efficacy levels. I conclude with an example of how one can apply an actor-oriented self-efficacy perspective to a study dealing with students’ self-efficacy related to proving.
Queer pedagogy has been underutilized in mathematics learning contexts, particularly at the undergraduate level, despite offering unique perspectives on gender, sexuality, knowledge, and normalcy that are valuable for scientific inquiry and which support queer students in STEM. In this paper I discuss queer pedagogy, present practical examples of its implementation in STEM courses at the undergraduate level and argue for its necessity as a tool for instruction.
Effective teaching relies on understanding and responding to students' mathematical thinking, enabling educators to build on student prior knowledge and deepen comprehension. While extensive research has established this link at the K-12 level, the research base for undergraduate mathematics education lacks a cohesive framework that integrates student thinking into instructional practices like planning and teaching. This paper addresses this gap by proposing a comprehensive framework designed to embed student thinking more deeply into undergraduate mathematics instruction, with the goal of improving teaching effectiveness and student learning outcomes. We conclude by discussing the implications for both instructional practice and research, as well as outlining the future directions for empirical validation.
Influenced by Piagetian constructivism and Vygotsky’s sociocultural theory, contemporary research has explored both individual cognitive development and the role of tools to mediate learning. In this theoretical report, we propose the Integrated Cognitive-Instrumental Model (ICIM), which synthesizes two prominent frameworks: APOS Theory (grounded in constructivism) and the Instrumental Approach (rooted in sociocultural theory). ICIM combines the strengths of both frameworks, where APOS Theory explains how learners move through stages in their cognitive development, and the Instrumental Approach shows how tools and artifacts shape and support this development. Together, these frameworks allow for a more comprehensive understanding of students’ cognitive development by bridging the gap between Piagetian and Vygotskian principles and highlighting the interplay between individual cognitive processes and the socio-cultural environment. ICIM offers a robust, multi-theoretical approach that provides a new perspective into student thinking with the potential to inform both research and instructional practices.
The growth of educational technologies outpaces our understanding of how to promote students’ mathematical development with such technologies. It has become increasingly important for researchers to problematize digital tool design so that educators seeking to enrich students’ mathematical knowledge with the use of digital tools support, and not hinder, student learning. In this theoretical paper, we draw on existing research on digital tools for mathematics learning to propose a framework that situates to-be-designed tools within a “design space”. We argue that an initial conceptualization of tools within this design space can help relate and distinguish tools from the literature that would be otherwise difficult to delineate or compare. We describe the spectra that comprise the digital tool design space, give examples of tools from the literature that fit within clusters of design space, and discuss the utility of such a framing.
We explore how one instructor used reflective prompts in a proof-based Elementary Number Theory course to add a writing component to the course. Students consistently reflected on their creative process and affective domain throughout the course. We share the design of the course, samples of the reflective prompts, and our deductive coding of them using two codes: creative process and affect. A preliminary examination of written responses from two students indicates that reflecting on their creative process had led them to be more engaged in the proving process in this course and other courses. They also reported that the readings and reflections helped them recognize and validate their mathematical actions, processes, and affective outcomes.
At the introduction to proof, Venn or Euler diagrams are commonly used to illustrate concepts in logic. Standard diagrams show 𝑃(𝑥) ∧ 𝑄(𝑥) as the intersection of truth sets for 𝑃(𝑥) and 𝑄(𝑥), and 𝑃(𝑥) ∨ 𝑄(𝑥) as their union. But textbooks rarely include a diagram for the conditional, and those that do use a diagram that differs from the standard form. Does this reflect typical mathematicians’ thinking? What do they draw when asked for a diagram to represent the conditional, and does this depend upon sentence formulation? This preliminary report describes a pilot study in which mathematicians and mathematics PhD students were asked to draw a diagram and provide a short accompanying explanation for either ‘if 𝑃(𝑥) then 𝑄(𝑥)’ or ‘𝑃(𝑥) ⇒ 𝑄(𝑥)’. Results show that they all draw essentially the same diagram, representing a true universal conditional. I illustrate variety in the provided explanations.
In this work, we review the literature on the teaching and learning of quantum mechanics situated in chemistry courses with the aim to provide a tool for education researchers and practitioners. We found that much of the research on quantum mechanics in the chemistry context involves students at the secondary level of instruction. Along with this, much of the research leans on mathematics and physics education research to support the claim that students have difficulties with mathematics; however, these claims require further investigation within chemistry education research.
In contrast to traditional homework sets that typically group problems by topics, an approach known as blocking, interleaving consists in alternating the topics for each problem set. Studies in middle school mathematics and college physics report important benefits of interleaving on students outcomes. The goal of the present study is to investigate whether similar benefits can be observed in the context of college calculus. Students in two sections of a Calculus for Life Sciences course received either blocked or interleaved weekly homework. On two surprise quizzes given in the middle and at the end of the quarter, the interleaved group outperformed the blocked group (d=0.18 and d=0.33 respectively). On a quiz given one month after the end of the course, the interleaved group outperformed the blocked group on the second half of the quarter but not on the first half. Limitations of these results are discussed.
Despite significant research efforts focused on the development of student-centered instruction, most mathematicians continue to teach through lecture. Consequently, researchers have called for further examination of the reasons why mathematicians appear to believe in their current teaching practices. Our study builds on previous research addressing this call by examining how mathematicians’ in- and out-of-class instructional decisions in upper-level, proof-based mathematics courses reflect their pedagogical goals. Four mathematicians participated in a series of three, one-to-one interviews examining their instructional design of a course they had recently taught. Data analysis is ongoing and is being guided by a deductive form of thematic analysis. In this paper, we provide an overview of literature on mathematicians’ pedagogical goals, describe the study design and plans for data analysis, and present preliminary findings in relation to one participant mathematician.
This study uses a survey to investigate mathematical convictions in tournament-style comparative judgements for written and visual proofs. The survey uses four theorems that can be proven using mathematical induction with each having four proofs: two written and visual. The visual proofs were equally convincing, and the respondents indicated that they believed these visual proofs to be mathematical proofs but not prove their theorems for all values n. The written proofs by mathematical induction were often the most convincing proofs—winning the overall tournament. Familiarity and content seemed to be prevailing factors for judging whether one proof was more convincing than another.
In this preliminary theoretical report, we problematize the language used in mathematical problems – in particular, with respect to the objectives and the directives of problems. We use this to conceptualize a set of eight problem types, described in three clusters, that run across mathematical domains. This paper explores the motivation for such work, as well as a rationale for why existing problem classifications in mathematics do not accomplish our intended aims; we identify potential applications of the framework, as well as challenges, for discussion.
In this study, we explore how students reasoned on function composition tasks with and without a dynamic graphical representation. Fifteen graduate students individually participated in the task-based and semi-structured interviews to sketch composite functions given their graphs, equations and dynamic graphical representations. One student’s responses were analyzed in this paper. Preliminary results suggest that: (1) incorporating multiple reasoning strategies, especially graphical reasoning, and (2) utilizing the dynamic graphical representation, helped build the student’s conceptual understanding of function composition.
Sense of belonging is an essential factor in supporting student success in undergraduate STEM courses. Female minoritized students, in particular, are less likely to report feeling a sense of belonging due to experiences of isolation, bias, racial and gender microaggressions, and stereotype threat. Although researchers have investigated sense of belonging at the campus or department level at PWIs, HBCUs, and research institutions, less is known about students’ mathematics classroom belonging at open-access, racially diverse institutions. In this report, I present the preliminary qualitative results from a mixed methods study regarding the aspects of classroom experiences that foster sense of belonging for Black and Latina female students in college algebra and precalculus classrooms at a Minority Serving Institution. My qualitative findings indicate that sense of belonging is influenced by mathematical microaffirmations, mathematical microaggressions and mathematics self-efficacy.
Diversity, Equity, and Inclusion (DEI) has become an increasingly important topic in mathematics education and in the professional development (PD) of Mathematics Graduate Teaching Assistants (MGTAs). Currently, little is known about the types of initiatives that are effective in supporting MGTAs’ use of equitable and inclusive teaching practices. Based on data from 45 MGTAs enrolled in our DEI-focused PD program, we propose a preliminary framework to elucidate the characteristics of activities that support MGTAs’ engagement in DEI issues. We illustrate the utility of the framework by highlighting examples of activities in alignment with the framework, as well as the memorable impact of these activities on the MGTAs who completed coursework in our PD program.
Series play key roles in calculus and other disciplines, yet research highlights several challenges with the topic, underscoring the need for curricular innovations to support conceptual understanding. In this paper, we introduce the Partial Sum Sequence Game and argue that it, with the guidance of a teacher-researcher, supported a pair of undergraduate students to make important connections between sequences, sequences of partial sums, and series convergence.
This research examines the ways students in a transition-to-proof class learn what a mathematical proof is and how to read, construct, and write proofs. We interviewed four university students who had successfully completed a transition-to-proof course about what experiences they thought most influenced their learning about proofs. We then interviewed two instructors of the course about their goals for student learning about the nature of proof and how they devised their instruction to advance those goals. We first describe the activities students engaged in and that the instructors provided for students that supported their apprenticeship into proof-based mathematics. We further analyze how students' characterization of these activities influenced their views of proof.
We report on a small-scale clinical interview study of students’ validation of AI-generated proofs. During these one-on-one interviews, students were tasked with using ChatGPT to generate proofs and validate those purported proofs. Consequently, neither the interviewer nor the participant knew if the justifications being validated constituted a proof before evaluating it. As such, no two participants validated the same proof. This required the development of novel post-interview analysis methods, which led to some valuable insights into students’ proof validation processes. We view the primary contributions of this work to be both the methodology we developed, and our insights gained into how AI-generated proofs provide a different lens for students’ validation and error detection.
The goal of this preliminary report is to demonstrate students engaging in transformational reasoning in an introduction to proof class. Utilizing transcript data and operationalizing Simon’s (1996) definition of transformational reasoning, these preliminary results include descriptions of transformational reasoning in an advanced math context, as well as how students used transformational reasoning in order to accomplish their mathematical goals.
Mathematics anxiety is detrimental to student achievement in math. This study uses a meta- analysis to investigate the effectiveness of various mathematics anxiety interventions on improving mathematics achievement of college students. We include intervention studies on college math classrooms with quantitative measures of both mathematics anxiety and achievement. Only RCTs and QEDs written in English are included. We use ERIC as a database. 8 studies were included in the extraction involving 800 participants. Studies generally indicated a moderately positive effect of math anxiety interventions on math achievement (p = 0.00124).
Criteria for proof are important because future teachers will need to evaluate students’ arguments as proofs or nonproofs. This preliminary report examines how a group of prospective and practicing teachers in a reasoning and proof course began to collectively construct criteria for proof. Given a set of hypothetical student arguments, their initial criteria differed, despite common mathematics coursework, including an introduction to proof course. Ongoing analysis uses documenting collective activity to understand how collective criteria and definition were constructed.
This paper describes a study of where and when calculus concepts and skills appear in the first semester of introductory physics. Following similar work in differential equations, engineering, and chemistry, we have adopted the Calculus Concept Framework and used the framework to identify which concepts and skills appear in each section of the first twelve chapters of a standard introductory physics textbook. Each section of each chapter was coded independently by three researchers and results were collated and discussed.
This phenomenological case study explores the metacognitive knowledge of undergraduate students that they accessed while constructing mathematical proofs, and the related actions they take in their proof process with that knowledge. Two pairs of undergraduate mathematics majors completed two proof-construction task-based video-recorded interviews and engaged in follow- up qualitative interviews asking them to watch clips of their work and reflect on their thinking. This preliminary report discusses the qualitative data analysis process that has explored the control actions taken related to metacognitive knowledge of one pair of students during the first proof task.
We present preliminary findings from an ongoing investigation into the discrepancies between how university-level mathematicians incorporate programming into their research versus their teaching. Motivated by earlier studies that highlight a gap in the use of programming between research and teaching, we explore this tension through interviews with mathematicians at a university recognized for supporting programming in science and mathematics teaching. Specifically, we draw on Engeström’s Cultural Historical Activity Theory to analyze the interview with one mathematician who illustrates this discrepancy. Our preliminary findings suggest connections to more deeply held epistemic beliefs about which tools and purposes are considered legitimate in mathematical practice.
We present a preliminary local instructional theory of students’ guided reinvention of equivalence classes and equivalence relations that has resulted from engaging in one iteration of the design research cycle. We leverage the instructional design theory of Realistic Mathematics Education (Freudenthal, 1991) to design a task sequence in an experientially real setting that can guide students to reinvent equivalence classes, equivalence relations, and their properties. In this paper, we present our revised task design describing the rationale, task implementation in a classroom setting, and analyses of student reasoning. Further, we conclude by discussing our plan to reimplement the tasks in another iteration of the design research cycle.
In all levels of mathematics, especially beginning at the undergraduate level, structural thinking is pivotal for deeper understanding or more complex reasoning about the relevant mathematical concepts. In this preliminary research study, we investigated the structural thinking of one advanced undergraduate mathematics student as he conjectured the definitions of various concepts in graph theory. We observed various affordances and obstacles to his structural thinking. Affordances included his attention to a general structural property and his use of analogical reasoning with corresponding structures in group theory and point-set topology. Obstacles included utilizing a constrained example space and relying on his familiarity with previously encountered mathematical concepts.
Mathematical literacy is a critical component of teaching mathematics as its own language. In this paper, we share the use of a writing to learn mathematics (WTLM) activity to help college students communicate their mathematical thinking and understanding. We conducted an embedded single-case study with two sections of undergraduate students enrolled in a Calculus II course focused on integral and multivariable calculus topics. Three WTLM activities were integrated into the course to provide opportunities for students to explain their symbolic, procedural, and conceptual understandings of partial derivatives and related topics. In this paper we share the framework and tools developed for this study and the data analysis plan and preliminary analysis steps.
Literature around the reading of proof has focused on experts and the strategies they use to understand the texts they are reading (Fang & Chapman, 2020; Paul, 2018) or in comparing students to experts in what the reader focuses on as they read using eye-tracking technology (e.g., Inglis & Alcock, 2012). Few have investigated the students and their strategies (Weber, 2015). In this preliminary report, we describe the most frequently seen actions taken by participants during various proof comprehension tasks. Our findings indicate that students use a variety of possible strategies to make sense of proofs they are reading. Some of these possible strategies are found to be used by mathematicians and secondary mathematics teachers, while other possible strategies are found to be used in general reading practices.
This preliminary report examines the reactions of mathematics instructors in Networked Improvement Communities as they engaged with student experience data to inform critical transformations within mathematics. Data-driven decision-making is promoted for improving equity; however, without a specific critical focus such efforts will likely fail. Examining data critically requires instructors to grapple with, at times, painful experiences of students, and their role in shaping these experiences. Using critical ethnography, we analyzed field notes to identify ways that instructors responded to data in meetings held from Spring 2023 to Summer 2024. Findings show that instructors' reactions to data often fall into two categories. Ego threat, characterized by reactions such as discomfort and defensiveness, effectively delegitimized student data. In contrast, some instructors exhibited curiosity and vulnerability, viewing students as experts of their experiences. Ongoing research will explore how ego threat shapes critical efforts and how fostering openness can enhance critical data-driven decision making.
Gatekeeper mathematics courses, including calculus, historically prevent marginalized students from pursing STEM majors by often focusing on procedural fluency over conceptual depth. Efforts to better prepare students for future career pathways include using tasks that emphasize authentic application contexts. In this study, we analyzed application tasks within the first half of the derivatives unit for two commonly used undergraduate Calculus I textbooks. Our analysis identified both the authenticity of the application context, and the conceptual understanding required to solve each task. By juxtaposing these task features, we found notable discrepancies between the degree to which tasks presented conceptual opportunities within authentic, real- world applications. These preliminary findings inform the need for further research investigating the relationship between conceptual understanding and authenticity in application tasks. This work can inform the design of novel tasks which appeal to both features, thus better preparing students to succeed in both calculus and future degree pathways.
Many studies have investigated the challenges students face during proof by mathematical induction (PMI). However, the majority of these studies are drawn from a small sample size. In order to generalize the findings from previous literature, we draw from a study of over 1000 proofs to develop an analytical framework for identifying common student challenges in PMI. In this preliminary analysis, we report our finding from analyzing 180 written responses for three types of PMI tasks.
The language we use to talk about, do, and learn mathematics informs how we socially construct our understandings thereof. This preliminary report discusses an instance from a different study where the use of feminine pronouns to describe smart logicians in a particular mathematical problem resulted in various reactions from participants. I analyzed participants’ interactions with the given task using an approach to discourse analysis that examines how the grammatical structures of language are used to construct particular meanings. Preliminary findings showcase how participants’ responses to the task reflect broader cultural understandings of who can be a smart logician. These findings are linked to the need for representation in curricular materials and the use of hat problems in the learning of logic and problem solving.
When learning about logarithms and solving logarithmic equations, students often misapply logarithmic identities. In some cases, misapplying an identity leads to the deletion of solutions or the introduction of extraneous solutions. This results from applying an identity without first checking that all logarithms in that identity are defined on the same domain. In this preliminary work, we examined 14 websites that College Algebra students might use when learning how to solve equations using logarithms, looking for mentions of the domains of definition whenever identities were stated or applied. Of the 11 websites that stated a logarithmic identity, only 4 mentioned domains of definition. Of the 11 websites that showed an example logarithmic identity application, zero mentioned domains of definition. We hypothesize that this lack of emphasis on domains of definition in teaching and learning logarithmic identities may lead to confusion when working with logarithms and in mathematics as a whole.
While many community college (CC) students come to higher education with specific career or life goals, these aspirations may be forced to change if students are not able to get past the gatekeeper of developmental mathematics. Racially minoritized students are disproportionately tracked into developmental mathematics classes upon entering CC, and often become trapped in a financial and emotional cycle of take-fail-repeat, with student debt linked back to developmental coursework totaling over $1.3 billion per year across the U.S. (Jimenez, Sargrad, Morales, & Thompson, 2016). This report presents a subset of data from a larger narrative inquiry project that focuses on humanizing stories from an intimate group of CC students, uncovering how they came to know themselves as learners and doers of mathematics. Findings provide insight into how mathematics learning experiences can function to dehumanize, while also paving the way to creating more humanizing undergraduate mathematics learning spaces.
Research has shown that self-assessment techniques can improve understanding and academic performance for mathematics students. As proof construction, validation, and comprehension have proven to be facets of mathematics with which students can use more learning opportunities, Mejía-Ramos et al. (2012) present an assessment model for proof comprehension. For our project, this framework is utilized in understanding proof self-assessment, with the goal of aiding students in the process of proof construction and validation. In this preliminary report, we provide results from the first phase of our project, in which we surveyed seven university mathematics faculty about their approaches to proof construction and self-assessment. Results from this study are being used to generate a proof self-assessment tool through the lens of proof comprehension that undergraduate students may use to improve their proof writing strategies. Pedagogical suggestions are also provided based on these results.
While extensive research has been conducted on curriculum development, teaching strategies, and student thinking in inquiry-oriented instruction, there is little focus on the nature of students’ inquiry as it relates to question-posing; we know almost nothing about how and why students ask questions in these courses. Since inquiry-oriented mathematics teaching aims to shift mathematical authority to be shared between teacher and students, we wonder about the role of student inquiry in this shift. We present preliminary findings from an inductive analysis of students’ questions and the mathematical authority relations that arise during classroom interactions where students ask questions in an inquiry-oriented undergraduate abstract algebra class.
This preliminary study reports on a brief survey of students’ course experiences in introductory university calculus. In particular, we compared the responses of students who took precalculus and calculus at the same institution with those who took precalculus in high school or other settings. Findings highlight significant differences in students’ reported calculus experiences between the two groups of interest, which indicates areas for improved alignment of course structures in undergraduate precalculus and calculus, especially around academic support. Building on this initial study, the data are being used to inform efforts to improve instruction and curriculum of both precalculus and calculus courses at the research site, starting with a new pilot calculus course in Spring 2025.
This study explores how WeBWorK, an online homework system, influences student mathematical identity, with a special focus on students at a Native American-serving institution. Mathematical identity refers to beliefs, attitudes, and emotions toward mathematics, impacting motivation and learning. Through an online survey we gathered insight on student perceptions of WeBWorK's role in shaping four identity dimensions: self-efficacy, understanding, confidence, and persistence. Results reveal significant differences in student perceptions compared to prior studies, with survey participants showing lower engagement across these dimensions. Preliminary analysis suggests that WeBWorK may not equally support the mathematical identities of all students, particularly those from minority or first-generation backgrounds. This research contributes to understanding how web-based homework systems affect diverse student populations, providing insights for improving such systems to foster mathematical engagement and identity development.
Interval estimation has become one of the preferred means for communicating the statistical findings of a study. Students, and even experienced professionals, often misinterpret frequentist confidence intervals. As part of the broader study in which this work is situated, we developed an instrument to assess undergraduate students’ statistical literacy surrounding confidence intervals. We conducted think-aloud interviews with six students who had recently completed an introductory statistics course. In this work, we qualitatively capture the meanings that students construct about attributes of confidence intervals to inform the results from the instrument. We use grounded theory to model how students conceptualize attributes of confidence intervals and propose evidence of interactions between interrelated ways of thinking. For this paper, we focus on one student’s conceptualizations about the confidence level and error attributes of confidence intervals to illustrate our modeling process in detail.
We report on a qualitative case study of a college algebra student, Jamie, who exhibited multiple conceptions of coordinate systems (Lee et al., 2020) and forms of graph reasoning (Johnson et al., 2020) during a single task-based interview. Our analysis of Jamie’s graph sketches and interactions with digital tools during the interview highlighted connections between her quantitative reasoning and conceptions of graphs. Specifically, Jamie’s conception of a quantitative coordinate system was closely linked to her reasoning about changing attributes and their relationships, whereas her conception of a spatial coordinate system was closely linked to her reasoning about the physical motion within the dynamic situation. Jamie’s case speaks to the complexity of students’ quantitative reasoning and illustrates the interconnectedness of their conceptions of coordinate systems and graph reasoning.
As K-12 classrooms have become more culturally and linguistically diverse, it is important to prepare pre-service teachers (PSTs) to develop their knowledge of designing and implementing instruction based on equity-based mathematics teaching practices (EMTP). We analyzed 41 PSTs’ lesson plans designed in a middle school mathematics methods course from five semesters. A theoretical thematic analysis revealed which EMTP categories appeared in the lesson plans. Findings indicate that PSTs seem comfortable and familiar with some categories, but they need support with other categories. In addition, methods courses should focus on EMTP so that PSTs strike a balance between EMTP categories by engaging students in non-routine, high cognitive demand tasks and by supporting diverse learners in mathematics classrooms.
This study examines how students engage in collaborative reasoning while working through an inquiry-oriented task on null space in linear algebra. Three students participated in a series of five online group interviews, followed by an individual interview. The task sequence involved exploring subspaces and null spaces using pathways in a school map. We analyzed interactions and identified moments where students built on or connected to one another’s ideas, influenced each other's reasoning, or left space for others to contribute. Our findings suggest that students' collaborative reasoning evolved over time, with instances of adopting and leveraging peers' methods. We highlight the role of positioning in group dynamics and how students' contributions to collaborative reasoning influence their understanding of mathematical concepts. The conclusions emphasize the importance of collaborative learning environments in facilitating deeper mathematical engagement and reasoning.
Derivatives are often considered the central part of introductory calculus (Kidron, 2019). While there has been consideration of how non-mathematicians use the derivative differently than their mathematical colleagues (Dray et al., 2019), there has been no survey data to aggregate these differences over many respondents. This study begins to explore how such data can be gathered and analyzed using a survey generated by the researcher to determine the process-object conception of the derivative (Dray et al., 2019; Zandieh, 2000) favored by various post- secondary teachers. In its current form, the survey distributed showed few significant differences between favored process-object levels in the framework of Dray and colleagues (2019). Survey improvements are discussed.
Contemporary news media frequently utilizes graphs to simplify the presentation of complex information. However, prevalence of misleading graphs on public platforms often makes readers susceptible to false information. This raises pertinent questions regarding the efficacy of our education system in adequately preparing students to interpret graphs. In this paper we present a case study that aims to explore undergraduate students’ interaction with misleading graphs and unpack their mathematical reasoning as they do so to identify patterns that can explain their graph interpretation abilities. The study's findings suggest that students who focused on the data in a graph, rather than its shape, demonstrated better proficiency in graph comprehension and in identifying misleading elements.
Through a scoping literature review of 40 RUME studies from 2019 to 2024, we examined how researchers use and describe mixed methods. We categorized these studies by country, subject area, and self-reported implementation process (e.g., sequential, parallel), methodological priority (qualitative or quantitative),and study design (exploratory, explanatory, triangulation). While mixed methods are used in RUME, we found that only half of the studies explicitly define their methodological framework, often missing opportunities to maximize the potential of integrating multiple data sources. Our findings highlight the importance of adopting more nuanced and deliberate mixed methods approaches to fully exploit the strengths of both qualitative and quantitative research in advancing the field.
Considerable attention has been given to the role of Mathematics Graduate Teaching Assistants (MGTAs) in fostering classroom norms. This is especially pertinent in active learning settings where students extensively engage in group work with minimal direct instruction. Social stereotypes can be pervasive within small group dynamics, often hindering participation of students in mathematical discourse. Implementing explicit classroom norms is an effective strategy to promote a more inclusive distribution of learning opportunities. However, more research is needed on how these norms are enacted and supported. Our findings reveal significant differences between MGTAs who consistently frame group norms and those who do not, highlighting the benefits of explicit norm framing for promoting meaningful discussions, structuring group work, and ensuring inclusivity. These results underscore the importance of professional development focused on norm setting to enhance inclusive student engagement in mathematics classrooms.
First-generation college students’ (FGCS) success in higher education is a growing area of research in the United States (US), but is infrequently recognized in the Norwegian education system; there is a need to investigate how FGCS are successfully navigating first-year mathematics courses to best support FGCS’ pursuit of a Science, Technology, Engineering, or Mathematics (STEM) degrees in the US and Norway. This study analyzes six interviews from Norwegian FGCS enrolled in first-year mathematics courses for navigational strategies employed to support their success in mathematics. The preliminary results highlight that the structure of the mathematics courses and the accessibility of resources at the university supported a community of collaboration between peers and instructors. The culture of teamwork across students positively influenced FGCS' success in first-year mathematics courses. We highlight the perspectives of each participant and provide directions for future research to gain a further holistic understanding of supporting FGCS thriving in STEM fields.
The flipped model of instruction has been championed as a method for increasing student engagement and enabling instructors to facilitate more active learning within the classroom. Research on the flipped model also suggests that it may also lead to better course outcomes than the traditional lecture approach. Given this potential for student success and engagement, faculty at California State University, Fullerton have adopted the model in many of their courses. In this preliminary report, we measure the impact of the flipped model on course outcomes of a Pre-Calculus course during the Spring 2024 semester. In addition, we compare student perceptions of the course elements that supported their learning between students enrolled in the flipped sections of the course and those in the non-flipped sections of the course.
Undergraduate math tutors frequently interact with students who are currently at a point of impasse or who have experienced failure and are seeking help. This affords tutors the opportunity to impact students’ understanding of mathematics as well as redirect their study efforts. Helping students accurately classify their mistakes is a worthwhile effort, but is challenging and requires training and support. However, to provide this training for tutors, we need to know how tutors view student errors. This report details preliminary results from a survey given to current tutors, asking them to classify authentic student mistakes on a test as “simple” or “not simple.” We discuss multiple reasons tutors give for their classifications, including some which stand in contrast to those we might use as researchers.
Drawing on the concept of “moves” from English for Specific Purposes (ESP) genre theory, we identified pedagogical proof moves—moves proof authors make for pedagogical purposes—in a corpus of 40 abstract algebra proofs. We share 12 such pedagogical proof moves, presenting three in greater detail.
This preliminary research report examines how undergraduates develop quantitative meanings of accumulation from rate, a foundation for understanding the definite integral and the Fundamental Theorem of Calculus. This study is a pilot study for a larger project. It consists of a pre-/post-assessment and a teaching experiment. The task sequence provides opportunities for students to develop productive meanings for speed, distance, and their relationship as rate of change/accumulation. Preliminary analysis of the pre-assessment indicates that students struggle to conceptualize average and instantaneous speed, which suggests that the teaching experiment can be productive. In this report, I outline the theoretical framework, task sequence, preliminary findings, and ongoing work. I conclude with areas for audience discussion: Seeking feedback on analysis and ideas for expanding upon this pilot study.
While Latine/Hispanic higher education enrollment rates have increased over time, enrollment in STEM, and mathematics in particular, has been slower to change. As part of a larger study, our team interrogates the experiences of those who have successfully navigated the mathematics pathway. Specifically, we look to understand how our research participants leveraged the support of both social (familial, peer, etc.) and institutional (mentors, faculty, administrators) along their academic pathways. We investigate primarily using in-depth qualitative interviews of five individuals who hold bachelor's degrees in mathematics and STEM PhDs, to ascertain both positive and negative influences on their education, utilizing the community cultural wealth framework. While the Latine PhD population is, of course, not monolithic, our findings suggest that social, familial, and navigational capital were instrumental in their ultimately successful navigation of both graduate and undergraduate education.
This study explores the evolving approaches of eight foundational math course coordinators, uncovering key insights into their coordination strategies and mechanisms to enhance their efforts. These coordinators oversee critical courses, including College Algebra, Quantitative Reasoning, Introductory Statistics, Math for Architecture and Construction Management, Precalculus, Calculus, and mathematics courses for prospective elementary teachers. Through a dataset derived from surveys, self-reflections, and professional development workshops, we investigated their perspectives and experiences as coordinators. We analyzed data from both the coordinators and the graduate student instructors they oversee. Specifically, we highlight the integration of instructional routines that promote mathematical reasoning and the development of course-specific dynamic calendar systems, both of which have the potential to improve the instructional effectiveness and coordination of foundational math courses. Our findings offer fresh perspectives on how to better support course coordinators in their crucial role, ultimately benefiting both instructors and students.
Professional learning communities (PLCs) can be a way to support college mathematics instructors in shifting their instruction to be more student-centered. This study focuses on what College Algebra instructors noticed when observing their colleagues’ teaching in a video club setting within the PLC. We used the instructional tetrahedron to classify what instructors noticed according to the four vertices: Content, Students, Teacher, and Tools and Technology. Results highlight that instructors’ discussions shifted away from Tools and Technology after returning to in-person classes and that instructors’ discussions about Tools and Technology differed depending on the in person or online class context. We also found that instructors frequently noticed Teacher interactions, which points to their perceptions about the goals of the PLC - to provide feedback about teaching and to generate ideas about how to improve one’s own instruction. Finally, what instructors notice about their colleagues’ teaching might shed light on that instructors’ goals and beliefs about teaching.
Women in advanced mathematics continue to be underrepresented despite representational progress in K-16 mathematics. Existing literature has investigated women’s mathematical identities at the undergraduate level and revealed varying ways women organize their mathematical selves in response to the masculine context of mathematics. In this paper, I expand on the existing identity research about women in undergraduate mathematics to better understand the mathematical identities of women in doctoral mathematics. I discuss themes related to women’s mathematical identities and the cultural discourse of “Women in Math.”
This study explores undergraduate mathematics instructors' perspectives on the benefits of Learning Assistant (LA) programs. Interviews with seven faculty members highlight LAs' roles in providing student support, enhancing instruction, and promoting student-centered learning. Instructors also noted that LAs gain valuable teaching experience, build relationships with faculty, and grow in confidence. These findings align with the established LA program goals of improving undergraduate education and preparing future teachers. However, the LA program goals of engaging faculty with educational research and institutional change were less evident. Future research will examine how LA use aligns with or diverges from established program objectives.
This paper presents preliminary findings on underrepresented students’ experiences as they navigate the mathematics research community. Three undergraduate mathematics majors from underrepresented groups were interviewed regarding their experiences at their home institution, research experiences for undergraduates, and an undergraduate mathematics conference. Using communities of practice as a theoretical framework, we explore these students’ status as legitimate peripheral participants in the mathematics research community of practice. Analysis led to bridging the communities of practice framework with Gutierrez’s (2009) four dimension of equity. In this paper, we focus on the ways in which students’ real or perceived legitimacy were impacted by identity and power. In particular, students experienced de-legitimization via the power of the dominant group and via less peripheral members.
Much research within mathematics education is dedicated to understanding the challenges that students often face when encountering new mathematical ideas. Some challenges, called epistemological obstacles (EOs), persist despite research-based instruction designed to target those challenges. The purpose of this report is to describe a methodology for mapping connections between EOs as they emerge during mathematical discussion and problem-solving. We illustrate this methodology through a preliminary analysis of whole-class data collected in an introductory proofs course.
Motivated by the goal of reducing equity gaps in achievement at a predominately white, public land grant research university, a team of mathematicians designed and implemented research- based strategies for promoting equitable instruction in a coordinated multi-section general education terminal mathematics course. To assess the impact of the course on students’ mathematics identity, we utilized a pre- and post-survey adapted from an instrument developed by Hazari et al. (2020) which included scales to measure the constructs of sense of belonging, interest, recognition, and utility/competence. Preliminary analysis revealed that the means of student responses changed in the desired direction from pre- to post-test, with sense of belonging showing the greatest, statistically significant change. Initial quantitative and qualitative findings suggest the instruction tasks described here, which are relatively simple and inexpensive to implement, can have a significant impact on improving mathematical identity for students in a general education mathematics course.
This preliminary report explores how students experience playful mathematics tasks in sections of a large Calculus I class. We implemented tasks that have been established to foster playful mathematical engagement and learning in 5 sections, and examined whether students experienced playfulness across multiple dimensions. We found that students enjoyed the class, felt engaged, and considered themselves to have understood the content, which they rated as at least somewhat challenging. These findings are being used to design professional development for future Calculus I teaching assistants, in order to examine potential scalability of playful mathematics and learning to lecture-based classrooms.
We focus on equitable and inclusive teaching practices in a professional development program (PD) for mathematics graduate teaching assistants (MGTAs). In doing so, we hope that MGTAs will grow to improve the experiences of their students. We analyzed over 1200 undergraduate surveys corresponding to nine MGTAs in our PD program. We completed statistical analyses by comparing means in two ways: (1) the aggregate data for each term and (2) for the individual MGTAs for each term. In the first analysis, we found a statistically significant improvement in scores from the first term to the last term of an academic year. However, for the second analysis, when we analyzed the means of each MGTA’s survey responses, we found that two of nine MGTAs had decreased scores over the course of the academic year. Ongoing research is investigating the extent to which experiences improved for different subpopulations of students within MGTAs’ classrooms.
Research suggests that a sense of belonging in mathematics can influence students' decisions to persist in STEM fields. Programs such as summer bridge programs and Louis Stokes Alliances for Minority Participation (LSAMP) programs are often utilized to promote STEM retention by helping students adapt to college STEM environments. This study explores the mathematics sense of belonging and identity of students in STEM while participating in an LSAMP program after their participation in a summer bridge STEM program. Survey and interview data from the students will be analyzed to identify how participation in STEM retention initiatives may have influenced their mathematics sense of belonging and identity. After taking their first summer mathematics course, students' overall sense of belonging in mathematics appears to decline, according to initial findings. Results of this study also suggest that having interpersonal relationships allowed one student to have an increased sense of belonging in mathematics.
Feedback—the process of leaving remarks, questions, or markings on student work with the aim of enhancing student learning—is one of the more instructionally powerful but least understood features in instructional design (Cohen, 1985). Past studies offer inconsistent results on the features of feedback and little is known about how some of these factors benefit students’ interactions with feedback and how deep a student engages in the revision process. This study aimed to learn more about both classroom perspectives of feedback between the instructor and the student. The main population included entry-level college mathematics instructors and students; we examined their feedback practices to gain a better understanding of what elements of feedback are being used and if they match the type of feedback that students report is most valuable. In this paper, we will only focus on the instructor study as that was the primary research question. Instructors graded and left feedback on a collection of student work samples while students completed a questionnaire that asked how they use feedback, how highly they value it, and which feedback type they find most useful. A thematic analysis was conducted on each comment and were classified into specific groupings. Results indicate that the instructor feedback was overall non-actionable but by a small margin. Students preferred actionable feedback, signifying a disconnect between the two populations.
Research in cognitive psychology identifies chunking as one of the driving information- processing mechanisms in human cognition. Conceptual chunking, a term specific to Halford’s relational complexity theory, provides theoretical insights into what constitutes chunking. Specifically, conceptual chunking explains the chunk formation as the result of temporal suppression of relations between the items within the chunk. In this article, I present the analysis of two undergraduate students’ proofs from the relational complexity theoretical perspective. Preliminary results suggest that conceptual chunking can help describe students’ mathematics and explain some of their difficulties related to proving algebraic conjectures.
This preliminary report focuses on investigating the range of gestures students use to explain the concept of derivative in different contexts. Students were asked to explain their meaning of derivative and how they make sense of the formal limit definition of derivative. In addition to taking an embodied cognition perspective in our analysis, we also draw from Zandieh’s Derivative Framework to make meaning of the range of gestures we observed. For this preliminary report, we analyzed a video recording of one student during a 30-minute assessment interview for a first semester calculus course wherein he displayed over 40 different gestures. Our findings focus on five representational gestures that were salient in the student’s explanation of derivative. We discuss methodological implications about analyzing gesture alongside a cognitive framework of a mathematical concept, as well as how our findings connect with existing themes in research on mathematical gestures.
This study investigates mathematics graduate teaching assistants' (MGTAs) conceptions of equity based on activities from professional development (PD) focused on active learning, inclusivity, and equity. Using Gutierrez’s (2008; 2009) four dimensions of equity (access, achievement, identity, and power), we analyzed survey data collected from MGTAs. These findings led us to create three MGTA profiles: Resource Advocacy, Asset-Based Collaboration, and Resilient Problem-Solving. These findings inform future PD design to enhance engagement with equity issues in mathematics education.
This preliminary report shares an outcome from a summer professional development (PD) activity with university instructors. Instructors participated in four PD meetings, then immediately taught a five-day summer workshop using inquiry, working primarily with first- generation minoritized students. While instructor participants’ exit interviews of the project identified their experience in the summer PD as pivotal to their development, we know little of how students experienced the instructors’ teaching during the workshop. Our analysis focuses on two items from student post-workshop survey wherein students shared their feedback of their instructor and their experiences more broadly. This analysis allowed us to get a good sense of the instructors’ individual practices and revealed convergence in their practices. Pedagogically, instructors utilized groupwork and deemphasized direct instructions, while prioritizing students’ engagement in discussions and struggling through conceptual ideas. Relationally, instructors were responsive to students’ mathematical needs and created a respectful, safe, and welcoming classroom environment.
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In a recent study, Leyva et al. (2021) call for researchers to scrutinize “precalculus and calculus instructors’ consciousness of whiteness and patriarchy embedded in undergraduate mathematics education” (p. 28). Inspired by this call, we propose investigating precalculus and calculus instructors’ beliefs about the source of racial and gender differences in mathematics achievement outcomes using the attributions of mathematical excellence (AME) framework (Jacobson et al., 2022). For this poster, we will describe the theoretical framework and present a study design which seeks to address the question: How does the AME framework capture university mathematicians' attributional beliefs of mathematical excellence?
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Mathematical modeling is a complex and dynamic process that has been studied and represented with multiple modeling cycles (e.g., Bliss et al., 2014; Blum & Ferri, 2009). However, these cycles can oversimplify students’ movements through various phases of the modeling process. As a result, research is needed to explain how students may deviate from the proposed modeling cycles as they work, and the context surrounding these deviations. Therefore, we analyzed pre-service mathematics teachers’ (PSMTs) progression through a high school-level mathematical modeling task using modified Modeling Activity Diagrams (Ärlebäck, 2009), based on the Bliss et al. (2014) modeling cycle. In our poster, we identify common themes to describe the deviations across groups of PSMTs and examine the contexts which elicited the deviations. These insights can be used to help prepare PSMTs to teach mathematical modeling.
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Higher education institutions have become increasingly aware of the challenges facing first-year students entering science, technology, engineering, and mathematics (STEM) fields, many of whom have inadequate mathematical preparation. Beyond these academic difficulties, students in STEM are highly vulnerable to mental health issues and are less likely to seek counseling services compared to their peers in non-STEM fields (Kalkbrenner et al., 2022). Recognizing these issues, Blinded State University developed an innovative approach to supporting first-year STEM majors through a collaboration between the Mathematics and Counseling departments. This poster shares the design of the Approaches to College Mathematics course, a research-based prerequisite course for Precalculus with integrated counseling support for struggling students.
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Feminist pedagogy humanizes learning for students by recognizing individuals as unique contributors within the social, cultural, and intellectual contexts. While feminism is readily used in subjects like humanities, less emphasis is placed in science, technology, engineering, and mathematics (STEM). This study aims to highlight feminism in mathematics through seeking to understand how feminist principals can be incorporated into mathematical modeling tasks. With a focus on Data Feminism and Reframed Problem Contexts, two mathematical modeling tasks will be designed and assessed for alignment with the aforementioned theories. This will lead to discussion of what characterizes tasks as “feminist modeling”, with an evaluation between the advantages and disadvantages of “feminist” and “non-feminist” modeling tasks. Findings will illustrate the feasibility and benefits of incorporating “feminist modeling” in mathematics education. This could potentially pave a new path for designing and implementing mathematical modeling curricular that considers students lived experiences, emotion, and embodiment.
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This study addressed the persistent gender disparity in STEM fields, particularly in engineering and robotics, by developing a survey instrument, “Enrolment in Engineering and Robotics Sensing Scale (EERSS).” The results of the study demonstrated good psychometric properties for the EERS scale, supporting the interpretation and use of the scale for understanding and measuring factors that influence the participation of female students in STEM fields.
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This study reports on how students experience peer learning in a first-year engineering mathematics course with portfolio assessment. The course has recently changed the assessment form from a high stake, closed book exam at the end of the semester, to a portfolio exam. We use the Community of Practice (CoP) framework to analyze students peer learning. The data is the midterm course evaluation, and the analysis incorporates mixed approaches. We find that the students reports that peer learning is particularly valued for its contributions to understanding complex concepts, maintaining motivation, and building a collaborative learning culture. Observations suggest that students naturally form Communities of Practice because of the digital tests, using these interactions to enhance their learning outcomes.
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In this poster presentation, we present pilot data describing how secondary mathematics teachers enrolled in a fully online, Real Analysis course in the Southeastern United States engaged with interactive digital activities. Further, we describe what mathematical meanings were elicited in this context during semi-structured interviews where students engaged with mathematical tasks via the activities. These pilot data will inform two future goals. First, it will inform the design of interactive digital activities in our Advanced Geometry course. Second, it will inform our future work on a larger study about overall engagement with various touchpoints (i.e., ways to participate) in fully asynchronous classes.
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In recent years, artificial intelligence (AI) has increasingly supported how students think about mathematics, especially with the rise of generative AI tools like ChatGPT. This growth requires educators to understand AI’s capabilities and effectively leverage them in teaching. This study focuses on using generative AI for undergraduate mathematics task design, exploring its affordances and constraints. The researchers employed a self-study approach, involving practice, analysis, and critical reflections, using ChatGPT in task design. Four dialogues were analyzed, each addressing a distinct application: mathematical exploration of singular value decomposition, generating coding in Python and Matlab for linear algebra, synthesizing frameworks for task design, and creating starting points for advanced topics like Lie groups. The study highlights both the opportunities and limitations of using AI in this context, aiming to inform broader task design practices and integration of AI into undergraduate mathematics education.
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I use a poststructuralist stance in this study to interrogate some values and norms for mathematical proof. While poststructuralist theory has been used in mathematics education, it has not yet been widely used to explore questions related to the practices of mathematical proof. In this study, I engage in an unstructured interview session with an undergraduate student, Sherine. I present Sherine with an excerpt about mathematical values and norms for proof along with two versions of a proof for the same theorem. Here, I share some of Sherine’s reactions to the reading excerpt and the two proofs. This study has implications for how we present proofs to students who are learning about them for the first time.
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An Intersectional, Quantitative Analysis of Undergraduate Students’ Responses to Racialized and Gendered Instructional Events in Undergraduate Calculus
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"This report details a teaching experiment (Steffe & Thompson, 2000) which consisted of clinical interviews (Clement, 2000) and exploratory teaching interviews (Sellers, 2020) designed to perturb selected students’ incoherent thinking. In short, this study addressed the following research questions: 1) In what ways are students’ understandings of the chain rule coherent with one another? 2) How do students respond to proof-texts which mirror their own incoherent thinking?"
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Extensive research has shown that a vital component of student success in mathematics classes is the ability to self-regulate (e.g., Bol et al., 2016; Isaacson & Fujita, 2006). Self-regulated learning (SRL) is the ability of a student to take an active part in their learning, by setting goals and motivating themselves, using different strategies to complete their learning task, and by reflecting on their learning and strategy usage (Zimmerman, 2002). This project looks at how students self-regulate their learning in developmental mathematics classes that use inquiry-based learning in the class and ALEKS software as a supplement, in order to answer the questions: (1) Is there a relationship between the intensity of IBL and students’ self-regulated learning skills? (2) Are there differences in the intensity of IBL across sections?. Five different sections were studied, where the classroom practices were measured with the Toolkit for Assessing Mathematics Instruction – Observation Protocol (TAMI-OP; Hayward et al., 2018), and the Inquiry-Oriented Instructional Measure (IOIM; Kuster et al., 2019), while SRL skills were measured with an adapted Motivated Strategies for Learning Questionnaire (MSLQ; Pintrich & De Groot, 1990). Some initial results are presented, such as a wide variability of implementation of IBL, and conflicting TAMI-OP, IOIM, and SRL skill scores.
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During pre-course-enrollment mathematics advising, incoming first-year undergraduate students meet one-on-one with a member of the mathematics department. Students with similar math backgrounds are not always recommended to take the same math courses. Despite having similar math backgrounds, students can end up with quite distinct course recommendations that vary widely in terms of their mathematical ambitiousness. This poster reports on this phenomenon and explores how gender mediates this advisor-student interaction and ultimately students' access to ambitious math coursework and persistence in math-intensive majors.
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Women and people of color continue to be underrepresented in mathematics. Thus, mathematics graduate teaching assistants’ (GTAs’) classroom experiences are often dominated by individuals who are white and/or male. This sends a problematic message about who can access and engage with mathematics. To better understand their educational experiences, we studied a group of mathematics GTAs’ reflections about short biographies from mathematicians of varying backgrounds. Analysis showed that reading such biographies provided validation for lived experiences, highlighted the lack of representation in mathematics, and shed light on ways that inequities in the education system continue to impact students of color. In particular, we discuss the ways that “Locks,” or a need in the field that is related to social justice or equity, show up in the GTAs reflections.
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