RUME 25
2023
Omaha, Nebraska
Number of pages: 1399
Number of papers: 212
- Samuel Cook
- Brian Katz
- Deborah Moore-Russo
2023
Omaha, Nebraska
Number of pages: 1399
Number of papers: 212
In order to overcome difficulties many students in mathematics and computer science face with rigorous proofs, some higher education institutes present a “Mathematical Reasoning” or “Transition-to-Proof” course. In this paper, we analyzed data from the exams of 310 students in such a course, focusing on the way they approach a proof regarding set inclusion, and whether they use a particular “proof framework” that we taught them when doing so. We found that the students are able to use the proof framework, and that using it is beneficial to them, but that a surprisingly large percentage of them fail to use it consistently.
In this descriptive case study, we explored how a mathematics educator integrated embodiment into a first semester abstract algebra course. We found that, in addition to gesture, the instructor encouraged students to interact with physical materials and simulate mathematics using their bodies. Our results offer practical implications by illustrating examples of how embodiment can be incorporated in an abstract algebra classroom.
The use of digital formative assessment in higher education mathematics is increasing throughout the world. Digital tools come with features such as endless ways of creating tasks and assessing in different ways. However, there is a need of research that focuses on what students do when working with digital tasks and how immediate feedback influence their learning process. This study examines students working styles and immediate feedback in STACK (The System for Teaching and Assessment using a Computer Algebra Kernel). The study reports on interviews with five first-year pre-service teachers who had completed a STACK-test for the first time, in order to get information regarding their working styles and perception of immediate feedback. The results indicate that students have issues regarding mathematical representations in a digital environment and a preference for line-by-line solutions in the feedback.
Sameness is a topic threaded throughout all levels of mathematics and yet it has not received much focus from the mathematics education research community. Our work categorizes the dimensions of variation that students highlighted when asked to describe sameness using a recent framework on sameness in mathematics. We captured students’ ideas about sameness through a pair of open response surveys at multiple doctoral-granting institutions across the United States. In this presentation, we report on the variation found in the student surveys and also compare our findings to a prior study of surveyed mathematicians.
While trends over the last few decades show increases in women and non-white scholars in STEM fields, there remains disparate attrition at different transition points within academia. The underrepresentation of certain demographic groups in STEM is broadly understood as rooted in systematic historical oppression and exclusion which continue their presence in various ways – but it is not clear how understood this is by faculty at large. To better understand faculty beliefs related to the (dis)advantagement of race-gender groups in STEM, we conducted a Latent Class Analysis on 945 responses from instructors of introductory chemistry, physics, and calculus courses at a variety of institutions across the United States. We then looked to see how those views relate to faculty support of policies and activities which might advance diversity and equity in postsecondary STEM education. Four clear patterns are discussed.
Although developing proof proficiency is an integral part of undergraduate mathematics, little is known about instructor grading practices and feedback conventions in undergraduate courses in which proof is taught. In recent years, small-scale studies have begun to shed light on how individual instructors at single institutions assign points and provide feedback on proof-related activities. Our study extends this work by describing the results of a large-scale online survey of mathematics educators (n = 86) from multiple institutions. Participants were asked to score and give feedback to hypothetical student submissions of a proof-adjacent assignment; we investigate participants’ responses and provide a taxonomy of their feedback.
Given the challenge of visualizing the main constructs of two-variable functions and their differential and integral calculus, it is important to consider the use and perceived potential of resources to contribute to students’ understanding of multivariable calculus. This case study considers how four instructors attempt to utilize resources in their multivariable calculus teaching and their motivations to do so. We study how these instructors think about the digital and non-digital resources that they use to foment students’ understanding. We also look at their reporting of the ways that instructors, students, and resources interact in multivariable calculus to determine if resource use is meant to facilitate visualization, reasoning, or communication. With this, the study proposes to contribute to the discussion of resource use in multivariable calculus.
In this contributed report I present results from a teaching experiment broadly focused on students utilizing machine-based computing as a mediating tool to learn fundamental concepts related to set theory and logic. Specifically, I highlight the mathematical activity of one student and his unique approach to finding the intersection of three sets. To understand how this student leveraged the computer programming environment, I used the analytical framework known as instrumental genesis, which can be utilized to investigate the confluence of an artifact (often a piece of technology) and the human mind to solve a mathematical problem. Results indicate that the student was able to use computational tools such as For Loops and the len(), or “length,” function as artifacts in his solution to finding the set intersection.
Increasing the participation and achievement of students in science, technology, engineering, and mathematics (STEM) in PK-16 education continues to be a focal area of educational transformation and research. Faculty members at Institutions of Higher Education (IHE) plan, implement, and investigate how program structures can support the development, retention, and overall success of undergraduate students in STEM. Active learning classrooms, especially in mathematics, are one way IHE are reforming student learning experiences, and also provide a unique opportunity to engage undergraduate learning assistants with faculty to support near-peer students and deepen their own learning. This report examines the role of learning assistantships in developing undergraduate preservice teachers' identity while participating in a Noyce Scholarship program. Findings from this study show that preservice teacher participants gained competence in a variety of forms and developed their identity as a teacher through their experience as learning assistants in active learning classrooms.
Research suggests that support offered by an instructor can significantly impact student experiences and that there are different types of support that instructors can offer (e.g., emotional support, informational support, etc.). This case study examines the goals and beliefs of a first-year Mathematics Graduate Teaching Assistant (MGTA) in an attempt to explain their in- the-moment classroom decisions about offering different kinds of support to their students. This study found that the MGTA’s beliefs about the nature of mathematics and how mathematics problems are solved were critical in understanding their tendency to offer a particular type of support. Understanding the ways that MGTAs support their students can initiate a valuable discussion surrounding novice mathematics instructors and their professional development.
As part of an effort to examine students’ mathematical sensemaking (MSM) in a spins-first quantum mechanics (QM) course, students were asked to construct an eigenvalue equation (EE) for a one-dimensional position operator. Sherin’s symbolic forms were used in analysis. The data suggest three symbolic forms for an EE, all sharing a single symbol template but with unique conceptual schemata: a transformation which reproduces the original, an operation taking a measurement of state, and a statement about the potential results of measurement. These findings corroborate prior literature on a construction task rather than a comparison or deconstruction task, and with a continuous variable after instruction on discrete variables.
Theoretical accounts of mathematicians’ disciplinary practice draw similarities to mathematical play, in that it can entail agency and autonomy, open exploration, creativity, imagination, and enjoyment. We analyzed task-based clinical interviews of 13 mathematicians to determine whether their problem-solving activity was playful and found evidence of three aspects of playful math: agency in exploration or goal accomplishment, self-selection of mathematical goals, and a state of immersion, investment, or enjoyment. We present these findings and introduce a form of problem-solving activity, the Explore-Focus Cycle, as one characteristic of playful math.
Mathematical play is hypothesized to foster student dispositions that mirror authentic disciplinary engagement, such as exploring, conjecturing, and strategizing. However, most research on mathematical play investigates the mathematics that can emerge from children’s natural play or play in informal spaces. We introduce the term “playful math” to highlight the potential of playifying school mathematics tasks and investigate both students’ and mathematicians’ problem-solving activity through this lens. Drawing on teaching experiments and mathematician interviews, we found evidence of playful engagement in both populations and identified similar activity structures across both groups, which we call Explore-Focus Cycles. We present two such cycles and discuss implications of playful math for student activity.
Research has noted that some aspects of proof remain implicit in undergraduate instruction. Accordingly, we propose that direct interaction with the rules of proving (e.g., deductive reasoning, proof-writing conventions) is of didactical value to students transitioning to proof- based mathematics. Using the commognitive framework, we aim to delineate the rules that newcomers to proof find challenging. These were solicited with a scriptwriting task that asked students to create a fictional dialogue about a proof-related mistake that they had experienced. The task was embedded in a homework assignment in an undergraduate course for academically motivated high-schoolers. Thematic analysis of 61 scripts resulted in a range of rules that concerned the use of examples, logical circularity, and proof layout. We discuss these findings in relation to students’ instruction and the existing literature on proof.
Given the crucial role that the graduate teaching assistants (TAs) play in undergraduate mathematics education, this study investigated how TAs interpret students’ work and plan to address students’ work in teaching. We analyzed TAs’ interpretations and plans according to approaches towards addressing students’ misconceptions discussed in the literature: viewing them as simply incorrect regardless of students’ thinking behind them, viewing them as a reflection of students’ flawed ideas to be confronted and replaced, and viewing them as resources for future learning. Our analysis shows that while TAs’ interpretations were often sufficiently rich to support using student misconceptions as resources for future learning, none of their plans used them in this way. Rather, most TAs’ plans were aligned with the confront-and-replace approach. Our results suggest that professional development could be designed to help TAs convert their rich interpretations into plans for using student misconceptions as resources for future learning.
As has been well-documented, the epistemological obstacles associated with teaching and learning mathematical proofs persist despite research-based instruction. We describe the ongoing design process of our NSF-funded project aimed at understanding and addressing those obstacles in introductory proofs courses, using proof by mathematical induction as an anchor. Our process is framed by two cycles of designed-based research. The first cycle corresponds to designing and implementing research-based instruction on mathematical induction, whereas the second cycle broadens the scope of our research to other introductory proofs topics. This paper reports on the outcomes of the first cycle, the transition between the first and second cycles, and the project's end products.
Students learn about function composition, (𝑔 ∘ 𝑓)(𝑥), in secondary school. From two given equations, one might identify the composite function algebraically via substitution, 𝑔(𝑓(𝑥)). But what about when functions are given as graphs? This study aims to explore how students were able to reason geometrically about composing functions. We identify common types of resultant graphs participants generated in trying to sketch the composition of various graphically- depicted functions, and the common types of reasoning we infer those participants engaged in while doing so. The results generate both questions, and hypotheses, about how best to support students’ development of deep – and diverse – meanings for composing functions.
Teaching observations can be used in multiple ways to describe and assess instruction. We addressed the challenge of measuring instructional change with observational protocols, data that often do not lend themselves easily to statistical comparisons. We first grouped 790 mathematics classes using Latent Profile Analysis and found four reliable categories of classes. Based on the grouping we proposed a proportional measure called Proportion Non-Didactic Lecture (PND). The measure is the proportion of interactive to lecture classes for each instructor. The PND worked in simple hypothesis tests but lacked some statistical power due to possible scaler ceiling effects. The measure correlated highly with a dependent measure derived from the Reformed Teaching Observation Protocol (RTOP), a holistic observational measure. The PND also provided effective descriptions and visualizations of instructional approaches and how these changed from pre to post.
Proof-writing is a core practice of mathematicians and a crucial skill every future teacher and mathematics major needs to acquire. A proof-based geometry course is a fruitful ground for developing proof-writing skills. Despite recent pedagogical and technological advances, proof remains challenging to learn and teach. We conducted a pilot study to explore potential advantages of an innovative digital proof platform, FullProof, which aims to advance students’ proof-writing skills in Euclidean Geometry. We integrated FullProof in three Geometry for Teachers (GeT) courses at three universities. We report on the findings pertaining to students’ interactions with FullProof, their perceptions of the usefulness of this tool to their proof-writing skills, and the lessons learned from the FullProof integration from an instructor perspective.
In broad terms, much of the research on graphical reasoning can be characterized as focusing on misconceptions, covariational and quantitative reasoning, and graphing as a social practice. In contrast, other research has focused on graphing as a cognitive process, emphasizing the fine- grained knowledge elements related to graphing, with a focus on characterizing ideas students associate with graphical patterns (i.e., graphical forms). This paper moves beyond graphical forms to characterize other categories of fine-grained knowledge – “graphical resources” – that are activated and used in concert when constructing and interpreting graphs. In this study, we identified six categories of graphical resources: graphical forms resources, framing resources, ontological resources, convention resources, quantitative resources, and function resources. We posit that holistically considering different categories of fine-grained graph-related knowledge resources can connect various bodies of research on graphing.
Intensive quantities result from quantitative operations on two or more extensive quantities. As such, their units of measure consist of “compound units.” Students regularly encounter symbolically-written compound unit structures that are directly given to them, rather than constructed or developed, such as m/s 2 , ft-lbs, or kg∙m/s. It is consequently important to understand how students might try to reason about such symbolically-presented compound unit structures, which is the focus of this study. We examined “ways of reasoning” students used to make sense of such units, and describe in this paper five themes that emerged during analysis: (1) decomposing into separate units, (2) treating units as variables, (3) using covariational/ multivariation reasoning, (4) posing a quantification, and (5) bringing in pure math concepts.
In this report, I discuss one potential role an applet plays in the context of a teaching experiment. Mathematics students are constantly tasked with understanding mathematical notation, which is often context-dependent or sometimes loosely defined. While the purpose of mathematical symbolization is to help mathematicians communicate concisely, researchers have indicated that students have difficulties understanding and utilizing mathematical notation to represent quantities. Therefore, examining the mechanisms by which students can refine their understandings into more conventional ones can be of value to the field of mathematics education. This study provides an example of using applets to help students engage in reflective discourse. The findings of this study indicate that giving students a medium to visualize their mathematics can support them in understanding what their mathematical symbolization actually represents.
Past research typically assumes that an instructor is a high school or college instructor, but not both. The mechanisms to obtain teaching credentials for each are also traditionally separate, but some instructors teach at both levels simultaneously or transition between them during their careers. To better understand them, we surveyed instructors with experience in both high school and college math teaching. For our qualitative study, we asked questions centered around the Pedagogical Content Knowledge domains within Mathematical Knowledge for Teaching (Ball et al., 2008; Shulman, 1986). In this paper, we discuss the survey, data collection, coding, and findings on teacher perceptions of their jobs that fall across institutional boundaries.
Despite the interrelated nature of math and science, limited work has characterized mathematicians’ and scientists’ beliefs about each other’s disciplines or courses. In this study, we use nine interviews with mathematicians and biologists to characterize the interrelationship of community values, methods, and accepted facts in math and science and their relationship to introductory courses. Results include similar descriptions of biology and of what scientists do by all participants, but differing characterizations of what math is and what mathematicians do and claims that math courses do not teach students what they need. Implications include a need for the mathematical community to better align introductory course material with their values so that people outside the community better understand what mathematicians value and do.
We present a study of how five early-career college math instructors using active learning describe their role as instructors and situate these descriptions in two existing theoretical frameworks: Herbst and Chazan’s theory of practical rationality and Louie’s culture of exclusion. This overlayed theoretical perspective, grounded in our case studies, points to levers for faculty professional development and instructional change.
In this report, I examine the capacity of fitness trackers to assist in differentiating students’ expe- riences of stress during classroom problem solving. I cluster N = 26 students’ heart rate vari- ability plots based on categories of description created using a phenomenographic analysis of their reported emotional experiences during a scripted lesson on roots of polynomial functions. My results reveal that fitness tracker data can be a valuable indicator of when a student will re- port classroom distress. However, the tracker data can give both “false-positive” and “false- negative” indications a student will report distress, reinforcing that biometric data cannot be treated as a proxy for students reported experiences of stress. Nonetheless, my results do support the use of fitness tracker data to guide deeper inquiry into student experiences. For example, fit- ness tracker data can assist in identifying classroom moments to discuss in a video stimulated recall interview.
Research indicates that investigating phenomena, rather than reproducing facts, should be a core experience in mathematics education. But despite its centrality to quality teacher education, it is still unclear how the effects of investigation tasks on teachers’ mathematical development can be analyzed. In this study, we demonstrate the potential of scripting tasks as an avenue for capturing mathematical activity during an investigation. Participants, comprised of prospective and practicing teachers, explored the concept of a “star polygon” and recorded their deliberations, ideas, and conclusions as a scripted dialogue. We analyzed their submissions using the construct of advancing mathematical activity (Rasmussen et al., 2005; Rasmussen et al., 2015), focusing on how participants’ interpretation and creation of symbols intersects other types of activity.
Authors of this proposal are members of an inter-institutional working group focused on the teaching and learning of transformations in college geometry courses taken by prospective secondary teachers. After exploring axioms and definitions for transformational geometry in our courses, we decided to shift to identifying not just what, but how students were learning about transformations in our courses. To explore this, we began a lesson study (Boyce et al., 2021). In this report, we discuss our engagement in the lesson study, its outcomes, and new directions.
In this paper, we seek to understand the formation of a national network constructed for change in STEM education. To center the role of making connections in community and collaborations between network members, we use social network analysis (SNA) methods. We examine the impact of an in-person conference with the goals to bring together disparate groups around a common issue on network collaboration structure. We also examine the effect of post-conference virtual community building activities on network collaboration structure. A major limitation of SNA is response rate which, due to the pandemic, artificially drove a loss of collaboration. However, this loss was mitigated by new collaborations, particularly by influencers and boundary spanners. In addition, a sustained, year-long suite of virtual activities designed for community building and collaboration increased the collaboration activity in the network by 12.5%.
This paper discusses findings from an ongoing study investigating mental mechanisms involved in the conceptualization of linear transformations from the perspective of APOS theory. Data reported in this paper came from 44-first year linear algebra students’ responses on a task regarding the range of a linear transformation. Our analysis revealed differences in APOS Levels, Stages, and Coordinated topics. Moreover, our findings pointed to connections between Levels/Stages of the range concept and mental mechanisms of representations of matrix multiplications. That is, our findings revealed that the absence of these representations from ones conceptualization process may result in irreversible misconceptions. More importantly, we identified more transitions occurring among Levels/Stages when linear combination expressions were considered as representations of matrix multiplications.
Professional development (PD) is often recommended to equip faculty to serve racially minoritized students through instruction. However, limited work has examined equity-oriented PD for mathematics faculty, who often hold views of instruction as race-neutral. This contributed report explores the influence of a two-year PD for faculty in a mathematics department engaged in equity-oriented reform at a Hispanic-Serving Institution. We present two cases of white faculty members who demonstrated a limited ability to interrogate their white racial identities in relation to their instructional impact, despite their engagement in a sustained PD designed to promote racial equity. Implications are provided for equity-oriented PD for mathematics faculty.
Despite the promise of inquiry approaches, student resistance can occur because active learning opposes their expectations of classroom norms and responsibilities. This study reports on students' experiences in inquiry-based learning (IBL) Calculus supported by learning assistants (LAs). We used an iterative process adapted from Hine (2013) to make changes to our study across three semesters to improve student experiences in IBL Calculus. Survey data of 285 students across 7 sections of IBL Calculus was supported by our observations of IBL Calculus and weekly LA seminars with 12 LAs to determine changes needed each semester. Quantitative data were analyzed using descriptive statistics and z-tests, and qualitative data were open-coded to evaluate the effectiveness of changes. We report changes made to LA preparation and course structure to support LAs’ instructional practices and increase student buy-in to IBL Calculus. Over the three semesters, the overall student experience in IBL Calculus improved.
Although the number of multilingual international students is increasing in undergraduate mathematics classrooms, there is little research about this population at the collegiate level. To better serve this growing population, it is important to understand their experiences using multiple languages in their mathematics learning. In this study, I constructed a narrative of one multilingual international student, Jia. I used narrative inquiry as a research method and collected data from synchronous- and email interviews and observations. The narrative demonstrates how Jia used her Chinese and English and how she dealt with her challenges. The study found that engaging in groupwork was particularly challenging for her as a multilingual student. I focus on not presenting her multilingualness as a deficit feature and discuss how to mitigate challenges multilingual students can face in their mathematics learning.
In this report, we begin to explore the value of leveraging constructs from quantitative reasoning for informing scaffolding moves that would assist modelers in overcoming blockages to their mathematization of canonical real-world problems from a first course on differential equations. Our claims are based in qualitative retrospective analysis of a set of 7 design cycles used to develop a task trajectory and resulting learning environment responsive to STEM students’ mathematical reasoning during mathematical modeling. We present a set of four scaffolding moves contingent upon and responsive to participants’ in-the-moment quantitative reasoning that guided them towards a meaningful model for a predator-prey scenario.
In-person tutorials present a prime opportunity for students to work with others to engage their problem-solving and critical-thinking skills. We investigate the use of interactive tutorials in a large, second-year service mathematics course as part of a quasi-experiment whereby students could either attend in-person tutorials (for a participation mark) or complete the tutorials individually without attending the live sessions and submit solutions for credit. Our analyses of student data and self-efficacy measures suggest that consistent attendance at in-person tutorials may maintain and improve students’ self-efficacy. Median changes to self-efficacy measures for students who attended in-person tutorials were higher than those who completed tutorials individually without attending the live sessions. The difference between the changes to the Emotional Regulation aspect of tutorial self-efficacy was statistically significant (p = .031), highlighting the role in-person tutorials play in supporting students’ mathematical learning beyond academic outcomes.
In this paper, we report on interviews with mathematicians exploring the ways in which they think about the relationship between equality and equivalence. Given sometimes unclear and conflicting presentations of equality and equivalence in the literature, we are motivated to understand subtleties about how these constructs interact; doing so can have pedagogical implications, which we explore in this paper. We present three major themes that emerged from our analysis of the mathematicians’ discussions of equality: 1) that equality represents a well-specified equivalence relation, 2) that audience matters when specifying equivalence and equality, and 3) that context is imperative when discussing equivalence and equality.
While inquiry approaches have the potential to open up more equitable spaces in the classroom, research has shown this is not guaranteed. In this study we focus on the role that status can play in creating (in)equities during small group work. In particular, we compare one low-status student’s interaction with two high-status students in two small group episodes. We found that the two episodes differed in terms of participatory and relational equity and our results describe how status and certain patterns of interaction may account for this difference.
The Mathematical Inquiry Project engages mathematics faculty from all Oklahoma’s public institutions of higher education in collaborative learning and resource development guided by three components of learning mathematics through inquiry: active learning, meaningful applications, and academic success skills. We report our analysis of participants’ interpretations of these components and the social and cognitive mechanisms they entail. Our results (1) provide a descriptive analysis of beliefs and priorities about learning mathematics through inquiry held by post-secondary mathematics faculty and (2) contribute to research and outreach on faculty change using a model incorporating implicit and explicit learning theories, within which the professional activity and collaboration develop a community that supports productive aspects of learning mathematics.
Secondary teachers and college instructors are required to take several graduate-level mathematics courses to advance in their careers. However, there is a common disconnect between graduate mathematics courses and the teaching of secondary mathematics. We developed a graduate-level mathematics for teachers course that was designed to help teachers connect abstract algebra, real analysis, and statistics to the content and teaching of secondary/ early college mathematics. In this study, we examined changes in graduate students’ confidence and perceptions of the relevance of advanced mathematics to teaching secondary and college mathematics. Students attributed changes in their perceptions of the connections between advanced and secondary mathematics to the course’s implementation of pedagogical scenario tasks. We provide implications for the mathematical preparation of teachers.
With increased availability of technology and data, the mathematical and statistical needs for many academic disciplines have likely changed over the past twenty years. In order to create appropriate general education courses in the mathematical sciences to meet these needs, a survey was conducted of individuals representing various academic programs in Alabama to determine the mathematical needs of students in their programs. The survey results indicate that traditional college algebra courses are not needed outside of the disciplines that require calculus. The needs of the non-calculus-based programs include linear and exponential functions; mathematical practices; statistical literacy; and the use of spreadsheets for data analysis.
Student understanding of key concepts from Single Variable Calculus and the role of infinitesimals was investigated. Analysis was done using a framework which highlights quantitative reasoning. Data showed robust understanding of instantaneous rate and change across multiple representations. I examined the rich student connections between those concepts expressed discretely in Algebra and expressed instantaneously in Calculus, and how conceptions of infinitesimal quantities were formed in order to support those connections. Implications for instruction are considered.
Learning to prove mathematical propositions is a cornerstone of mathematics as a discipline (de Villiers, 1990). While the field has generated research that has analyzed the final products of proof (Selden & Selden, 2009) and there are frameworks for analyzing problem-solving processes (e.g., Carlson & Bloom, 2005; Schoenfeld, 1992), much remains to be known about analyzing undergraduate students’ proving processes. With a focus on impasses in the proving processes, this study provides a more fine-grained account by characterizing students’ overall proving process and navigating actions. The analysis lays the foundation for discussing the relative productivity processes students may take as they are engaging in proving.
This paper explores how two undergraduate students come to perceive exponential structure in a series of word problems. This study investigates whether features of task situations can create opportunities for supporting students in recognizing situations as appropriate for using exponential models. The analysis is guided by coordination classes theory (diSessa & Sherin, 1998) and Transfer-in-Pieces perspective (Wagner, 2006; 2010), which emphasizes the role of multiple instantiations of a concept across multiple contexts as learners construct their understanding. The results suggest that contexts that embody the notion of recursive dependency may help students activate and use appropriate knowledge resources related to exponential models.
This report contributes novel insights into how undergraduate students think and reason about the number of tiles within a given tiling of the plane. We juxtapose two theoretical perspectives— units coordination and spatial-temporal-enactive (or S*-) structuring—to provide an analytic account of how two undergraduate students reasoned about the tiling. In particular, this work suggests potential qualitative differences in how undergraduate students might engage in multiplicative reasoning in an unfamiliar spatial context.
Students use symbolic manipulations when performing cancellation as they solve equations. Rather than performing rote manipulation of symbols, we want students to understand the meanings that these symbols convey. In this paper, we examine three pre-service mathematics teachers’ symbolic forms for additive, multiplicative, and compositional cancellation. Students used symbol templates where they either wrote both the identity and inverses, one but not the other, or neither. We further explore students’ varied conceptual schemas associated with these templates, as well as the relationships that exist among these symbolic forms.
We discuss two Dynamic Geometry Software applets designed as part of an Inquiry-Oriented instructional unit on determinants and share students' generalizations based on using the applet. Using the instructional design theory of Realistic Mathematics Education, our team developed a task sequence supporting students' guided reinvention of determinants. This unit leverages students' understanding of matrix transformations as distortion of space to meaningfully connect determinants to the transformation as the signed multiplicative change in area that objects in the domain undergo from the linear transformation. The applets are intended to provide students with feedback to help connect changes in the matrix to changes in the visualization of the linear transformation and, so, to changes in the determinant. Critically, the materials ask students to make generalizations while reflecting on their experiences using the applets. We discuss patterns among these generalizations and implications they have on the applets' design.
Developing a rich understanding of linear combinations is key to understanding linear algebra. In this paper, I explore the rich connections students make between the geometric and numeric representations of linear combinations through playing and analyzing a video game. I look at population of students who have never taken linear algebra before and analyze how they structure space using the video game, Vector Unknown, as a realistic starting point. I detail and analyze this activity including the activities that transition them from 2D to 3D space.
Those who lead the preparation and assessment of novice college mathematics instructors for teaching (Providers) do their work in many ways (e.g., course coordination, seminars, workshops). Using data from a large national survey, this study examined reporting among 95 Providers about the structures of their departments, their goals for the professional development work they do, and their relative valuation among goals. Respondents completed a sorting and ranking activity about professional development goals and answered an open-ended question describing their sorting decisions. Qualitative coding identified six main themes for the respondents’ 285 descriptions. Quantitative analysis used the rankings of goals within respondents’ sorting categories to examine how Providers describe and value professional development goals related to professional community, classroom and department culture, and instructor response to students within their classrooms.
Professional development about teaching is an important lever for change toward evidence- based instructional practices. Yet few studies in higher education offer robust evidence about whether and how instructors’ teaching practices actually change as a result of professional development. We studied instructors who attended a four-day intensive workshop on inquiry- based learning (IBL) in college mathematics and who provided data about their teaching before and after. Of 293 instructors who attended, 136 provided complete survey data and 15 video- recorded their teaching, which we coded using two observation protocols. Comparing pre- workshop and follow-up teaching data, we find significant and well-corroborated changes toward IBL methods across all three methods, with effect sizes near 1, generally viewed as large in education research. The findings demonstrate that well-designed professional development can have a substantial impact on instruction even within the first year of implementation.
As part of an effort to examine student understanding of expressions for probability in an upper-division spins-first quantum mechanics (QM) context, clinical think-aloud interviews were conducted with students following relevant instruction. Students were given various tasks to showcase their conceptual understanding of the mathematics and physics underpinning these expressions. The symbolic forms framework was used as an analytical lens. Various symbol templates and conceptual schemata were identified, in Dirac and function notations, with multiple schemata paired with different templates. The overlapping linking suggests that defining strict template-schema pairs may not be feasible or productive for studying student interpretations of expressions for probability in upper-division QM courses.
With an increased call to attend to diversity, more postsecondary mathematics instructors are interested in using equitable and inclusive teaching (EIT) practices. Yet, ambiguity persists on what EIT practices look like in mathematics classrooms. Our study investigates the characterizations of EIT from 13 participants, each involved in a year-long equity focused workshop, with the purpose of describing what topics arose in their characterizations. Our analysis shows characterizations of EIT included both the student and teacher role in EIT, and focused heavily on awareness of students’ experiences and backgrounds, student engagement, and teacher self-reflection. We provide specific details from participants’ characterizations for each category and connect them to established EIT literature. These findings can help researchers and department leaders engaged in change for EIT see what topics in EIT are being consumed and what topics might need emphasis or introduction.
Weinberger and Fischer (2006) designed a framework for analyzing learning the context of asynchronous discussion activities. Operationalizing this framework, we analyzed the social and cognitive aspects of a discussion activity in an asynchronous calculus course. From this analysis, we identified aspects of Weinberger and Fischer’s (2006) framework lacking in explanatory power for mathematics-specific discourse and developed an amended framework. We propose that this amended framework may enable in-depth analysis of major dimensions of students’ mathematics knowledge construction as they engage in activities in an online asynchronous modality. This framework may also support the curriculum development for online asynchronous mathematics coursework.
During the COVID-19 pandemic, citizens have had to make new decisions about vaccination and social contact based on their personal risk estimates and the risk estimates of experts and governments. We analyzed survey data with 50 participants and interview data with three participants to model how South Korean citizens quantify risks associated with COVID-19 vaccination and infection and the severity and level of concern they associated with these percentages. We found participants’ quantifications for COVID-19 risks and their meanings for their estimated risks are substantially different than experts. Participants used benchmarks such as 10% risk and 50% risk to estimate and reason about their COVID-19 risks.
Classroom community plays a pivotal role in student success, especially when students have the opportunity to interact with their peers and instructors. This study explores how students perceive classroom community and its impact on their learning. We present findings from an essay reflection assignment where students were asked to share their experiences with learning mathematics. We found that students felt more comfortable to ask questions and were able to develop interpersonal skills in conjunction with mathematical skills because of the classroom community. Conclusions drawn from this research can provide insights for instructors who are interested in fostering strong classroom communities in engaging learning environments.
Using data from teaching experiments and theories from quantitative reasoning, we built second-order accounts of students’ mathematics on how they conceived change via operating on existing quantities. We report three different ways STEM undergraduates structurally conceived change as they constructed mathematical models for real-world scenarios.
The use of computer programming is ubiquitous for research mathematicians, and two common practices are facilitation of experimentation and testing of conjectures. These practices align with empirical re-conceptualization, which is the process where empirical data is used to identify patterns, form related conjectures, and then re-interpret the conjectures from a structural perspective. However, literature on empirical re-conceptualization has only examined student work done by hand. We present case study data of one student, Allen, using the programming language Python to facilitate empirical re-conceptualization as he found the closed-form solution for binomial coefficient C(n,k). We discuss why the computational environment was productive for empirical re-conceptualization and provide avenues for future research.
More college mathematics classrooms are adopting active learning practices like collaborative learning. Participation in these practices typically requires students to engage in verbal communication and interpersonal interactions, which are mediated by language. Despite this, the experiences of multilingual students have often been overlooked in the literature on active learning. Through analyzing interviews with 26 multilingual students, this study explores what participation in active learning meant to these students. It also explores the resource that students perceived to be necessary in order to participate in active learning, for example, being comfortable speaking English. Finally, this study demonstrates how classroom discourses and broader social discourses about language shaped students’ perceptions about participation.
The Orchestrating Discussion Around Project (ODAP) was conceived of as an investigation into how instructors might support a student-centered classroom with aims of engaging students in authentic mathematical proof activity. The design-based research focused on engineering tasks and specific high leverage teaching practices to promote student engagement with three focal proofs in Abstract Algebra. In this report, we share a series of project learnings related to supporting students in accessing formal proof activity and promoting more equal participation amongst students. We focus specifically on how complex task launch, structuring group work, and working with student ideas in whole class discussion may be shaped to support more students in authentic mathematical proof activity.
Standards-based grading is an alternative assessment method that is gaining traction in undergraduate mathematics courses. Reported benefits for students include reduced math and test anxiety, increased achievement rates, and deeper understanding of concepts. However, evidence is mixed, and researchers have called for qualitative studies to explore students’ feelings and experiences with standards-based grading. In this study, we examined 228 College Algebra students’ responses to an end-of-semester reflection assignment. We discovered that some students expressed a shift in their feelings and mindset towards mathematics, and that these shifts were connected to the standards-based grading assessment structure. We present these findings, which describe how students viewed their learning and how shifts in their feelings and mindsets related to specific features of the course. We also discuss the internal and external motivations for these shifts and pose questions for future research related to students’ experiences with non-traditional assessment structures.
Although research on active learning suggests strong connections to equitable and inclusive teaching, few studies have explored the relationship between the two concepts. For this proposal, we examined how 13 participants in an equity workshop described the relationship between active learning and equitable and inclusive teaching. We used set theory as an analytic tool to categorize their descriptions. We found that all participants saw active learning and equitable teaching as related. The two most commonly held views were that active learning and equitable teaching are related yet independent sets, or that equitable teaching is a subset of active learning. We discuss implications of these findings and pose questions for further exploration.
We analyzed end of course reflections of graduate students in mathematics courses for practicing teachers. Participants were asked questions about mathematics content, habits of mind, and whether being a concurrent mathematics teacher while taking the graduate course impacted the value that they saw in either area. Participants were also asked to reflect on whether they believed the course would have been valuable when they were an undergraduate student. We first coded these reflections with Hoffman and Even’s (2018, 2021) framework including essence, doing, and worth of mathematics. As a result of our analysis, we suggest extending their framework to mathematics and mathematics teaching, and to include content. Discussion includes suggestions for future research to understand how teaching mathematics reframes learning mathematics, and the importance of framing to understanding the uptake of mathematics courses for both prospective and practicing teachers.
In the past two decades, there has been a trend in materials for mathematics courses for prospective secondary teachers: more opportunities for teachers to “apply mathematics to teaching”. That is, materials increasingly highlight how mathematical knowledge learned in the course can be useful in secondary teaching, and provide opportunities for teachers to harness this knowledge in simulations of teaching. There is little known about the effects of this curricular reform on teachers’ competence. In this report, we use data from the Mathematics of Doing, Understanding, Learning, and Educating for Secondary Schools MODULE(S2) project to examine the potential impact of using such curricular materials. The data include over 300 prospective secondary teachers’ responses to 3 sets of Likert pre-/post-term surveys addressing: mathematical knowledge for teaching; expectancy for enacting selected core teaching practices; and valuing of enacting these practices. We found mean increases across the survey results. We conclude with directions for future research on the impact of this curricular reform.
Although it is well known that motivational and cognitive resources influence secondary teachers’ instructional quality, less is known about the tertiary instructional factors that influence secondary teachers’ development of these resources. To address this gap, we report on factors that prospective secondary teachers attribute to their learning. We draw on survey responses of 70 prospective secondary teachers enrolled in mathematics courses for teachers using Mathematics of Doing, Understanding, Learning, and Educating for Secondary Schools (MODULE(S2)) materials in one of four content areas. We triangulate response themes with data from 300 prospective secondary teachers on their perceptions of instructional practices used in a mathematics course for teachers using the same suite of materials. Then, we compare these themes with literature documenting implementation of mathematics curricula in these courses. We argue that coordinating mathematics content, applications of mathematics to teaching practices, and tertiary instructional practices are key to success of these mathematics courses.
Previous studies have established teacher questions as a significant interaction component in proof-based courses whether lecture-based or student-centered. However, little research has been conducted to investigate behind the types of questions asked in class. In this report, we share an analysis of three instructors’ lessons on the First Isomorphism Theorem. We characterized instructor question types and then conducted interviews with each instructor to investigate how their pedagogical beliefs relate to the types of questions they ask in class. We found that the instructors held similar pedagogical beliefs about student learning, but diverged in their views on the purpose of the course. Both beliefs on learning and their beliefs on purposes of the course linked to the nature of the questions asked in class.
The importance of student-centered approaches took on new importance when the COVID-19 pandemic caused a tectonic shift in college instruction in 2020. In Fall of 2021, I reached out to all 13 of the new graduate teaching associates (GTAs: graduate students who were instructor-of-record for undergraduate courses) in the department. Eight of the 13 completed three rounds of survey and two rounds of interview. The goal was to find out what they knew about student-centered instruction and how that changed over time. GTAs spoke of the importance of supporting students in evaluating knowledge claims, learning how to learn, how to collaborate, and how to seek help. Their greatest opportunities for professional growth at the end of the year were with teaching that supported students to: become capable of self-assessment, to be resilient, and in building skills in what to do when they (students) do not know what to do.
The mathematics education community has simultaneously experienced a call for mathematical creativity and the implementation of computing within the classroom. What has not been explored are the ways in which these two calls are complementary. This study investigates how computation, enacted through coding, can bring about mathematical creativity within a series of linear algebra computational notebooks designed using the Understanding by Design framework. Specifically, through the analysis of a group of students, this paper highlights how a computational prediction and reflection cycle enables exploration and student originality while also facilitating connections to multiple representations.
Advanced, proof-based mathematics courses typically include learning goals of developing understanding of the relevant content of the class. The content always includes definitions of concept in that they are foundational to theorems and proofs. This study explores what three students believe it means to understand a definition, the actions they take to develop that understanding, and the rationales they use to explain why their actions are productive. The students held divergent beliefs about what it means to understand a definition. While all cited lecture attendance and homework as critical activities in their learning, they diverged on the use of outside resources and the rationales for why they might or might not be useful.
Learning to interpret proofs is an important milepost in the maturity and development of students of higher mathematics. A key learning objective in proof-based courses is to discern whether a given proof is a valid justification of its underlying claim. In this study, we presented students with conditional statements and associated proofs and asked them to determine whether the proofs proved the statements and to explain their reasoning. Prior studies have found that inexperienced provers often accept the proof of a statement’s converse and reject proofs by contraposition, which are both erroneous determinations. Our study contributes to the literature by corroborating these findings and suggesting a connection between students’ reading comprehension and proof validation behaviors and their beliefs about mathematical proof and mathematical knowledge base.
We report results of a longitudinal case study of one Intermediate Algebra student Sierra’s experiences relearning factorization from 8th grade through Precalculus. This data comes from a larger project investigating six developmental mathematics students’ perceptions of the experience of relearning individual topics and reflections on relearning across two semesters of college algebra courses. Such a student-centered perspective is critical to developmental mathematics educators who are currently grappling with high rates of failure and attrition in such courses nation-wide. In examining Sierra’s learning experience as one of relearning, this report sheds light on how students may use their previous algebra learning experiences to make predictions about the kind of understanding they will come to have about particular topics, and to motivate particular behaviors and affective states during instruction. Furthermore, we illustrate how such topic-level experiences may be used to shape learning experiences in subsequent mathematics courses involving algebra.
This paper presents six categories of undergraduate student explanations and justifications regarding the question of whether a converse proof proves a conditional theorem. Two categories of explanation led students to judge that converse proofs cannot so prove, which is the normative interpretation. These judgments depended upon students spontaneously seeking uniform rules of proving across various theorems or assigning a direction to the theorems and proof. The other four categories of explanation led students to affirm that converse proofs prove. We emphasize the rationality of these non-normative explanations to suggest the need for further work to understand how we can help students understand the normative rules of logic.
Math anxiety negatively affects student learning and academic performance. Students with high math-anxiety exhibit physical, mental, and emotional symptoms. These symptoms often have a short-term and long-term impact on students’ mathematics learning and their performance both inside and outside of school. This study investigated the effects of inquiry-based learning (IBL) on Calculus I students’ math anxiety, compared to lecture-based instruction. The short version of the Mathematics Anxiety Rating Scale (MARS-S) was used as a pre- and post-test to identify the students whose anxiety from pre- to post-test—greatly increased, greatly decreased, and did not change much. The selected participants were interviewed one-on-one to understand their perceptions and experiences of learning Calculus I. The results showed that some activities, such as the opportunity to work in groups, optional and ungraded homework, and the instructor's welcoming, caring, and amicable nature decreased IBL students’ anxiety. On the other hand, the instructor’s readiness to explain the material in class when students asked him to do so and his care for student success decreased lecture-based students’ anxiety. However, the tests and exams and anticipating the instructor’s call for a response increased anxiety among both groups of students.
We report on findings from a mixed methods study on preservice elementary teacher learning outcomes as a result of being enrolled in courses where their instructors were participating in professional development designed for those new to teaching content courses for future teachers. The study contributes to the current sparse literature within the RUME community on student performance as a result of professional learning opportunities for mathematics faculty. Results indicate improved undergraduate learning outcomes among preservice teachers when instructors engaged in a professional short-course that both fostered growth of instructors’ own Mathematical Knowledge for Teaching (MKT) for teaching undergraduates and provided instructors with opportunities to learn about and support prospective teachers’ development of MKT (for teaching children).
The method of Least Square Approximation is an important topic in some linear algebra classes. Despite this, little is known about how students come to understand it, particularly in a Realistic Mathematics Education setting. Here, we report on how students used literal symbols and equations when solving a least squares problem in a travel scenario, as well as their reflections on the least squares equation in an open-ended written question. We found students used unknowns and parameters in a variety of ways. We highlight how their use of dot product equations can be helpful towards supporting their understanding of the least squares equation.
Mathematicians often use set-builder notation and set diagrams to define and show relationships between sets in proof-related courses. This paper describes various meanings that students might attribute to these representations. Our data consist of students’ initial attempts to create and interpret these representations during the first day of a paired teaching experiment. Our analysis revealed that neither student imputed or attributed our desired theoretical meanings to their diagrams or notation. We summarize our findings in two vignettes, one describing students’ attributed meanings to instructor-provided set-builder notation and the other describing students’ imputed meanings to their personally-created set diagrams to relate pairs of sets
We investigated seven undergraduate students’ definitions of two terms used in mathematics education research: “identity” and “math person.” Students’ interpretations of these terms were compared to an existing identity framework (Cass et al., 2011; Cribbs et al., 2015). Participants’ definitions for “identity” varied widely, emphasizing different aspects of personal, social, and mathematics identities. In contrast, students were fairly uniform in describing a “math person” as an individual who enjoys and/or is proficient in mathematics, which is consistent with known factors related to mathematics identity. This second finding bears relevance given the use of self- identification as a “math person” in measurements of students’ mathematics identity. These results guide our development of survey items meant to assess aspects of student affect in the mathematics classroom, with the ultimate goal of providing instructors actionable insights into their classroom climate.
This mixed-methods study examines the self-regulation strategies that first-semester undergraduates use in a first-semester calculus course through quantitative analysis of student survey responses and qualitative analysis of student interviews. The relationship between these self-regulation strategies and both mathematics identity and mathematics self-efficacy was found to be positively correlated. The strategies of metacognitive self-regulation, effort regulation, and time and study environment were found to be the most commonly used strategies by academically successful students based on a large-scale survey given to 188 first-semester calculus students as well as from interviews that occurred throughout the Fall 2021 semester.
In this paper, we describe the results of administering a survey we created to investigate students’ meanings for a universally quantified variable. Respondents with various levels of proof-course experience reviewed statement interpretations showcasing different meanings for quantified variables, indicating their preferred examples, which examples they believed to be viable, and a truth value for the statement. We developed the various interpretations from meanings described in our previous qualitative work on student quantification (e.g., Sellers et al., 2021). Our results suggest that students with a range of proof-course experience may benefit from reviewing diverse interpretations of a quantified statement. Still, students with less proof experience tended to accept more interpretations as viable or chose inappropriate truth values. In particular, we identify two non-normative interpretations most students believed were viable. We recommend proof-course instructors explicitly address these meanings in their classroom to foster student understanding of quantified statements.
This study seeks to explore and understand the challenges of teaching precalculus for the first time from the perspective of a novice Mathematics Graduate Student Instructor (MGSI). This qualitative multiple-case study followed three MGSIs through a semester of teaching to understand their needs and inform efforts to support instructors. Findings indicated novice MGSIs encountered many challenges including covering course content, estimating class time, facilitating activities, maintaining student engagement, and adjusting to variation in students’ preparation. However, balancing dual responsibilities of student and instructor, working with students’ negative responses to engaging teaching methods, and interpreting students’ performance on assessments emerged as the most difficult challenges to overcome.
This study explores how instructional interventions and teacher moves might support students’ learning of logic in mathematical contexts. We conducted an exploratory teaching experiment with a pair of undergraduate students to leverage set-based reasoning for proofs of conditional statements. The students initially displayed a lack of knowledge of contrapositive equivalence and converse independence in validating if a given proof-text proves a given theorem. However, they came to conceive of these logical principles as the teaching experiment progressed. We will discuss how our instructional interventions played a critical role in facilitating students’ joint reflection and modification of their reasoning about contrapositive equivalence and converse independence in reading proofs.
The purpose of this study was to investigate how students in a probability course interpret a theorem and how they apply the theorem when asked to determine whether two random variables are independent given the value of the variables’ covariance. We considered how responses of students with prior training in logic compare to other students, as the theorem is a conditional statement. We used a quasi-experimental design to investigate the impact of two instructional treatments, using isomorphic colloquial statements and using Euler diagrams. Participants largely gave conventional truth values for variants of the conditional statement prior to instruction, but many of them used the converse and/or did not use the contrapositive when given covariance values and asked to determine whether two variables were independent. Our results do not consistently show a significant difference in students’ choices based on their prior logic training or either treatment.
This study examined the tasks in the exercise sections of a popular open-source statistics textbook in order to characterize the types of reasoning the tasks are likely to elicit from readers. To achieve this, I examined the narrative sections of the book alongside the solved examples to determine the extent to which the kind of guidance provided was relevant in solving the tasks in the exercise sections of the book. Findings show that although most tasks could be solved by closely following the procedures and methods in the book (imitative tasks) there were other tasks that could elicit more nuanced forms of reasoning (creative mathematical reasoning tasks). These tasks were more prominent on the tail ends of the exercises section than on the beginning parts. Implications of these findings are discussed.
Recent state policies have pushed for more diversity in the STEM pipeline. One proposed solution is the California Assembly Bill (AB705) (California Community College, 2018), which requires students to take and pass gateway math courses within their first year at community colleges. In theory this policy seems like a solution, however pragmatically there is a lot to be learned about its implementation and consequences for students. In this paper, we address the research question: What are students experiences with placement and advising at a local two- year college. We conducted a descriptive qualitative case study (Yin, 2009) that draws upon s student outcome data and focus groups. We focus on 5 STEM majors’ experiences. Findings suggest most students passed their gateway math courses within a year, however students described negative placement and advising experiences: receiving deficit-oriented suggestions from counselors and limited counselor availability. We discuss implications in the paper.
Inquiry and active learning instructional methods have largely been regarded as equitable and beneficial for students. However, researchers have highlighted math classrooms as racialized and gendered spaces that can negatively impact marginalized students’ experiences in such spaces. In this study, I examine the development of one argument, and whose ideas are solicited and leveraged, in an inquiry-oriented linear algebra course with an eye toward participatory equity. I found that gender related most to the inequity of participation in argumentation and that only men participated in generalizing activity. This study adds to the growing literature addressing equity in inquiry and active learning math settings.
We analyze students’ cooperative small group work on an open-ended mathematical modelling task. Introduction of new ideas and techniques challenged traditional students’ views on mathematics and conventional solution routines. This caused tensions in the student activity system pointing towards primary and secondary contradictions. To identify contradictions, their discursive manifestations were used.
Function composition is highly useful for making sense of many fundamental ideas in calculus and related areas of mathematics. However, of the large body of research investigating students’ understandings of function, very little has been centrally focused on function composition. In an effort to contribute to the existing findings pertaining to function composition, this interview- based study involved an exploration of a first-semester calculus student’s use of function composition when solving unfamiliar problems. The student’s responses revealed potential evidence of his understanding of function composition, function as an action or process, and covariational reasoning.
In this report, I examine the recommendations made by researchers to use magnitude bars (or dynagraphs) with students to support their reasoning about quantities. More specifically, I look at implications of an undergraduate pre-service secondary mathematics teacher’s reasoning when using tasks with magnitude bars designed to support students’ meanings for geometric formulas in dynamic contexts. I illustrate the ways in which a student reasoned, ways that both resonated with researchers’ recommendations but also introduced unanticipated difficulties that resulted from the introduction of the magnitude bars. Specifically, I show two different ways of comparing magnitudes a student illustrated in the Painter Problem-using amounts of change reasoning and length comparison reasoning. The results of this study contribute to the literature on using magnitude bars to support students’ reasoning by illustrating the different ways in which a student can reason with magnitudes.
Linear algebra is an important topic for many STEM students and presents unique challenges for teaching and learning. In this study, we analyzed one mathematician’s instructional materials and spoken language. Our theoretical framework is based on Tall’s (2008) three worlds of mathematical thinking and Harel’s (2008) ways of thinking. The goal of the study was to examine the complexities of a mathematician’s ways of thinking while moving between the three worlds of mathematical thinking, while orchestrating the important aspects of linear combination, subspaces, and span.
The contributed report highlights linear algebra students’ understandings of the matrix representations of certain linear transformations defined on with respect to given bases in a DGS-MATLAB assisted pedagogical environment. The research focused on the diversity of strategies that students utilized specifically in the process of diagonalizing linear transformations in a series of in-depth qualitative interviews within a framework of modes of description and thinking in linear algebra (Dreyfus, Hillel & Sierpinska, 1998; Sierpinska, 2000). Data analysis revealed three main categories: (i) synthetic-geometric mode manifested as the visual alignment strategy in which students visualized each image vector as a multiple of the corresponding preimage vector; (ii) analytic arithmetic mode prevailed in the process of expressing the image vectors in terms of the preimage vectors; (ii) analytic structural mode was observed in the process of determining the diagonal form of the linear transformation with reference to the transition matrices and the commutative diagram.
This theoretical report argues that conditional inference is central to undergraduate mathematics, and that there is potential for using and adapting conditional inference tasks from cognitive psychology to study ways in which reasoning develops (or does not develop) with mathematical expertise. Specifically, developing mathematical expertise might improve conditional reasoning in mathematics only, or it might improve conditional reasoning in a way that transfers to abstract and/or everyday contexts; alternatively, it might develop conceptual understanding, so that people become better at conceptually valid mathematical inferences but more vulnerable to invalid responses typically seen in tasks with everyday content. This report argues that, to test and distinguish these possible developmental mechanisms, we need new conditional inference tasks with mathematical content. It also describes the first stage of a project designed to develop (and later use) such tasks, initial outcomes of which will be reported at the conference.
Communities of practice (CoPs) provide a structure that allows individuals with a common goal or purpose to come together to engage in collective learning (Wenger-Trayner & Wenger-Trayner, 2015). The COMmunity for Mathematics Inquiry in Teaching (COMMIT) Network was developed to support regional Math CoPs, called COMMITs, composed of undergraduate mathematics faculty interested in using active learning and inquiry teaching approaches in their courses. For the past three years, we have utilized the value framework, presented by Wenger et al. (2011), to better understand the layers of value faculty experience as they engage in COMMITs and the broader network. In doing so, we have identified modifications to the original model from Wenger et al. that may further interrogate and articulate the layers of value that can be used to help advance and sustain CoPs longterm. We present our theoretical adaptation in the following report.
Over the past thirty years, researchers have developed constructs to investigate teachers’ knowledge base and how it relates to their teaching. These constructs include Mathematical Knowledge for Teaching (MKT), Mathematical Meanings for Teaching (MMT), and Key Pedagogical Understandings (KPUs). In this report, I describe each of these constructs, their uses, and how they are related. I then describe a recently introduced construct, Ways of thinking about Teaching an Idea (WTTI) (Carlson, Bas-Ader, O’Bryan, & Rocha, in press) and illustrate the usefulness of this construct for investigating a teacher’s mathematical understanding of sine function and how this understanding is related to the teacher’s instructional goals and actions.
Recent research has explored the potential for students to create new mathematical concepts by analogy with previously known concepts. However, there is more to be understood about the intricate processes of analogical reasoning that students leverage when creating new concepts. In this paper, we describe an initial theory explicating the role of abstraction during analogical concept creation. First, we introduce the abstracted domain to operationalize abstraction during analogical reasoning. We then describe two mechanisms of analogical concept creation through which abstraction may occur while reasoning about a ring-theoretic analogue to subgroups: (1) comparative analogy, and (2) inductive analogy. To broaden the interpretive scope of the theory, we further apply the mechanisms to scenarios involving meta-analogical reasoning and pedagogy. Implications for research and teaching mathematics with analogy are discussed.
In this exploratory theoretical analysis, we apply Grice’s maxims for cooperative communication to make sense of the ways that proofs are written and understood. We argue that proving can sometimes be productively viewed as a form of cooperative communication. Grice’s maxims place expectations on the prover and provide norms for how proofs should be written. The reader’s expectation that the prover is adhering to these norms allow for the possibility of implicature—that is, the prover can imply statements to the reader that are neither explicitly said nor logically necessary. We use this analysis to better understand the practice of proving and to suggest a future line of empirical research.
Although “developmental math” is widely discussed in higher education circles, exactly what developmental math encompasses is often underdeveloped. In this theoretical report, we use a sample of highly cited works on developmental math to identify common characterizations of the term “developmental math” in the literature. We then interrogate and problematize each characterization, particularly in terms of whether they serve equity-related goals such as access to college credentials and math learning. We close by proposing an alternative characterization of developmental math and discuss the theoretical implications. We see this as a first step towards conversations about how developmental math could be conceptualized.
In this theoretical paper, our aim is to start a conversation about how “levels” in mathematics are operationalized and defined, with a specific focus on “college level”. We approach this from the lens of developmental stages, using this to propose an initial framework for describing how learners might progress along a developmental continuum delineated by the kinds of reasoning/justification, generalization/abstraction, and types of conceptions that they hold, rather than by the particular computations learners are able to do, or the kinds of mathematical objects with which learners are engaging.
In the context of proofs, researchers have distinguished between syntactic reasoning and semantic reasoning; however, this distinction has not been well-explored in areas of mathematics education below formal proof, where student reasoning and justification are also important. In this paper we draw on theories of cognitive load and syntactic versus semantic proof-production to explicate a definition for syntactic reasoning outside the context of formal proof, using illustrative examples from algebra.
In this theoretical paper, we describe how algebraic transformation could be reconceptualized as a process of substitution equivalence, and we discuss how this conceptualization affords mathematical justification of transformation processes. In particular, we describe a model which deconstructs the process of substitution equivalence into core subdomains which could be learned serially and then re-integrated, in order to make them accessible to students with lower prior knowledge in syntactic reasoning. Our aim in presenting this model is to start a conversation about what the core components of knowledge might be in order for students to reason about and justify algebraic transformation using symbolic representations.
This theoretical contribution draws on earlier work by Herbst and Chazan (2012; also Chazan et al., 2016) in which they describe the position of a mathematics teacher in an educational institution as accountable to stakeholders who issue four types of professional obligations. We propose an application and adaptation of that framework intended to address the case of instructors who teach undergraduate mathematics courses to future teachers. Considerations of not only the academic but also the professional ends of these courses are key in our application of the theory of obligations.
Multivariable and vector calculus (MVC) encompasses a wide array of different types of functions and the extension of numerous ideas from univariable calculus to MVC contexts. In this paper, I reflect on some of the more recent work on student thinking and learning of derivatives in MVC settings by leveraging some of the recent work on students’ multivariational reasoning and student thinking and learning of linear algebra topics. I conclude with examples of ways to conceptualize derivatives as linear transformations to support students in progressing towards a unified notion of the derivative concept throughout MVC instruction.
The idea of intellectual need (IN) has received much interest from instructors in trying to design tasks that engage students in impasse-driven learning. However, we argue that the literature on IN is currently insufficient for supporting the careful design and implementation of tasks meant to provoke IN. In this paper, we examine two shortcomings: (1) What exactly IN can be created for, and (2) How an instructor might support students in navigating the experience of resolving the confusion and constructing the targeted meanings. For the first of these, we describe the category error of thinking of producing IN for a “topic,” and use the idea of conceptual analysis to suggest a way to address this shortcoming. For the second, we bring in control-value theory to explain what an instructor might attend to in order to ensure that the disequilibrium stays productive and does not lead to frustration and disengagement.
Making progress in justice, equity, and diversity in post-secondary teaching and learning requires systemic change. The development of novice instructor professional knowledge is a critical subsystem of the undergraduate mathematics education system. Novices play key roles in instruction and have the potential to play key roles in change efforts later in their careers. Yet, there is little in the way of theory to support research and development in this area. In other fields, professional development that engages novices in building skill at self-sustaining, generative change as professionals is the ground in which agency for change is seeded and nurtured. We describe two dimensions of professional skills for interacting with ideas and people: decentering and interconnecting. In this report, we explore and illustrate the role of these dimensions in professional development for novice college mathematics instructors.
I propose a theoretical model for teaching, learning, and assessing mathematics learning in the undergraduate mathematics classroom combining participatory and acquisition views of learning. The model, Balanced Learning Needs Framework, was created from an organization of the 10 learning needs proposed by Sfard in 2003. I align the model with different forms of classroom instructional techniques used in an undergraduate Calculus I classroom. Included with the organization of the learning needs, this conceptualization has the potential to impact both teaching and assessment methods in undergraduate mathematics classrooms.
In response to enduring methodological tensions in analyzing collaborative discourse, we detail an assemblage of argumentation analytic methods that can support research about faculty learning communities interacting across substantive differences. Drawing on our research with a cross-institutional faculty online learning community, we use data to show how theories from discourse analysis, systemic functional linguistics, and argumentation modeling can be operationalized to support researchers in brooking methodological tensions, including framing argumentation as the topic of or a resource for investigation and considerations of collaborative discourse as both process and content. Our methodological findings illustrate an example of this operationalization, highlighting analysis of transdisciplinary, collaborative discourse in a community composed of instructors of college geometry courses required for pre-service teachers. We share possible uses for this methodological approach vis-a-vis research about the professional work of undergraduate mathematics education and pre-service teacher preparation.
In this study, mastery-based grading was implemented in three sections of precalculus taught at two institutions. We investigated student perceptions towards this grading system by analyzing math autobiographies written by the students at the beginning of the semester as well as responses to surveys that were distributed at three points in the semester. In this paper, we present our results and discuss the implications to help support students during mastery-based courses.
There has been a push to develop curricula that engages students in various mathematical practices (e.g., defining). As more curriculums of this type arise, there is a greater need for assessment tools for students participating in these practices. We see Modeling Eliciting Activities (MEAs) as one answer to this need. In this study we discuss how the six design principles of MEAs can be used to (re)design an assessment tool for students’ defining activity.
Research has shown that support from collaborative professional development communities benefits faculty introducing student-centered instruction to their classrooms (Henderson, Beach, & Finkelstein, 2011; Speer & Wagner, 2009). While the RUME community has some understanding of how professional development groups can be effectively formed to support teachers (e.g., Kelley & Johnson, 2022), further research is needed. In this study, I investigate the norms established within one such professional development community: an online working group designed to support teachers’ implementation of an inquiry oriented abstract algebra curriculum. Preliminary results indicate that participants established norms related to working on tasks, contributing ideas and responding to each other’s contributions, and sharing authority.
In this study, we explore the influence of an experiential learning social justice math class on undergraduate students’ beliefs about mathematics and mathematics teaching. Twenty-three undergraduate students participated in this ten-week long class. Students reflected on their learning by responding to weekly journal prompts and designed a learning product for the high schools. Students’ journal entries and project design will be analyzed to answer three research questions. How does the class influence students' beliefs about mathematics? How does the class influence students' perceptions about who does mathematics? How does the class influence students' understandings of how to teach math equitably? Preliminary results suggest the following: Students' views of mathematics expanded from a focus on application and problem solving to include intrinsic values of mathematics such as beauty and exploration. Students’ perception of mathematicians became more inclusive. Students’ understanding of equitable teaching varied.
Extant literature has emphasized the importance of education research being theory-based. To this end, many studies have an explicit “theoretical framework” section that describes the theoretical assumptions that inform the research. Nevertheless, there is large variation in the extent to which studies incorporate theory in the research process. This work describes a literature review conducted across discipline-based education research, focusing on the role of theory in research. We focus on a snapshot of theory use by analyzing studies published in 2021 (n=589) across biology, chemistry, engineering, mathematics, and physics education research journals. Preliminary analysis focuses on trends related to the presence of a theoretical framework and reflecting on how the theory is used in the research process. Our goal is not to evaluate the quality of research, but rather describe how theory is used to inform future work and open a cross-disciplinary dialogue about theory.
This article reports mathematical understanding of one of the precalculus students who participated in a teaching experiment. The experiment was designed to help students strengthen the understanding of the concept of rate of change and linear equations to prepare them for the calculus concepts: tangent line equation and linear approximation. Before the experiment, the student understood slope mostly as an algebraic ratio and could not use the slope to determine the y-intercept of the line. With the intervention, she strengthened her knowledge of slope as both algebraic and geometric ratios and used her knowledge to construct linear equations. In addition, she was able to use slope to determine linear approximations of function values in various functions. The results suggest that when provided with appropriate tasks and prompts, precalculus students can be better prepared for their upcoming calculus concepts.
Near-peer mentor models to support students in mathematics are becoming increasingly used at institutions across the United States. One model, the Learning Assistant model, has been implemented at Crossroads State University. Using Yosso’s Community Cultural Wealth framework, this paper discusses the qualitative impact this model has on students, learning assistants, and the instructor of a College Algebra class. Based on findings, additional qualitative research is suggested to further investigate the Learning Assistant model.
The Teaching TRIOS project aims to promote strength-based faculty peer-observation of teaching for professional development. In this project, we present preliminary work from a comparative case study of two teaching TRIOS groups, one largely strength-based and the other primarily weakness-based. We seek to understand the levels of reflection facilitated by a strength-based and weakness-based approach to observation through the lens of the Onion Model for levels of change (Korthagen, 2004). Preliminary results indicate that a strength-based approach may promote more frequent reflection upon deeper aspects of oneself, such as beliefs, identity, and mission.
We present a case study of a student engaging in “proof without claim” tasks. In these tasks, students are presented with proofs in which the proposition being proven (its claim) is removed. Students are asked to ascertain what this missing claim is. This brief report serves (1) to illustrate how our novel proof without claim tasks can provide insight into how students engage with proofs and (2) as an existence proof that some students conceive of proofs as packaging for calculational math problems.
Instructors who prepare prospective teachers should provide opportunities for them to make connections between Abstract Algebra and secondary mathematics. This study focuses on an instructor who guided prospective teachers to connect properties of algebraic structures with equation solving procedures by evoking their intellectual needs (Harel, 2013) for computation, structure, causality, and communication. The students resolved such intellectual needs by using ring properties as tools to justify their solutions to equations. We characterize the intellectual needs the instructor evoked, along with her pedagogical moves used to evoke them during class discussion. We discuss how we will extend this analysis and provide implications for teaching.
Eigentheory concepts are central in mathematics and physics; they serve multiple functions, such as symbolizing physical phenomena and facilitating mathematical computations. The words associated with eigentheory develop and vary over time (e.g., eigenvector, eigenstate), as do the associated symbols (e.g., Ax = λx, H|En> = En|En>). The focus of this study is how the concept of “eigen'' develops over time for a quantum mechanics classroom community. We analyze the public displays of form-function relations used in the classroom community (Saxe, 1999). In this preliminary report, we summarize the forms and functions identified in the first 19 class sessions and share progress on our analysis of shifts in the uses of forms and functions over time.
We report on experiences of two Calculus 1 instructors who implemented teaching practices geared towards fostering students’ mathematical creativity. While teaching a coordinated Calculus 1 course, these instructors participated in weekly professional development sessions on such teaching practices. Our preliminary analysis of participants’ experiences describes how they navigated between the coordinated course structure and the added teaching practices discussed in the professional development. We utilize a zone theory lens to gain insight into participants’ navigations in their professional environments, observing tensions between existing zone of proximal development and the zones of free movement and promoted action complex.
Mathematical microaggressions (MM) refer to the subtle ways in which mathematical authorities use language, behavior, and assumptions that communicate negative messages to students that they do not belong in mathematics (Su, 2015). This study analyzes the reflections of 173 undergraduate mathematics students who were asked to reflect on an article (Su, 2015) about MMs. Findings show that students experience different types of MMs, including microslights, microinsults, and environmental microaggressions. Our results indicate that female students were more likely to report experiences with MMs than male students. This research supports the need to investigate this phenomenon further and to develop initiatives at departmental and institutional levels to encourage more inclusive spaces in math classrooms.
A key feature of inquiry-based instruction is to develop formal mathematics by directly building on student thinking and contributions, which often requires engaging students in refining their work. In our experiences we have noticed that it is challenging for instructors to support this refining activity and so we see a need to better understand what instructors can do in-the- moment to initiate refinements. In this study, we investigated how three instructors across four courses guided their students through the refinement of one definition. We identified three different moves that these instructors used to support the definition refinements: (1) suggest an edit with implicit mathematical reasoning, (2) suggest an edit with explicit mathematical reasoning, and (3) ask a question to create a problematic situation. The first two moves focused on the refinement itself whereas the third move focused on a problem to motivate a need for a refinement. We discuss future research directions and implications of our work.
Student success is typically measured by grades in coursework and 6-year graduation rates (York, Gibson, & Rankin, 2015). The purpose of this exploratory data analysis is to develop statistical methods that will provide a deeper understanding of student success, specifically success in first-year mathematics (FYM) prerequisite courses as well as the STEM courses they serve. We consider the many possible student course-taking sequences (paths) as a complex network system. We model the paths that students take with a network of vertices (courses) and edges (links between courses) by using 10 years of registrar data to generate vertex and adjacency matrices, and associated graphical representations (e.g., heat maps and network graphs). We also will conduct relevant exploratory data analysis to determine the importance of certain mathematics courses and transitions to overall student success.
In this preliminary report, we share results from an ongoing project related to the professional development of K-12 math and science teachers in Washington state. The goal of this work is to establish and sustain a network of STEM teacher leaders who can support teachers statewide in using effective STEM pedagogy. In the project’s first phase, we surveyed 290 stakeholders, including administrators, support specialists, teachers, and other community/industry partners, in order to identify attributes valued in a STEM teacher leader. Initial findings from this mixed- methods study indicated that STEM stakeholders prioritize teacher leaders’ abilities to foster the incorporation of integrated, community-based STEM projects and culturally responsive pedagogy in theirs and their peers’ teaching. We seek input from the RUME community on the project’s second phase, in which we will study the efficacy of a particular change model in facilitating cross-institutional systemic change toward the adoption of STEM best practices.
Calculus is a pivotal course for postsecondary students and serves as a gatekeeper to advanced mathematics courses and STEM majors. This study considers the frequency with which instructors report engaging in practices that may help to strengthen introductory calculus students’ prior knowledge. Using survey responses (N=136), this study reports on results from a set of questions that measures these practices. Results illustrate that instructors may use various practices to address students' prior knowledge in their introductory calculus classroom. Additionally, instructors provided examples of practices they use in response to student prior knowledge that are not explicitly considered in the literature. This implies that a closer look at what is happening in the classrooms may be key in understanding the current state of how instructors are dealing with student prior knowledge in introductory calculus.
Research has highlighted that actively involving students during instruction can lead to positive outcomes for students. However, college mathematics instructors may need support to develop the knowledge and skills necessary to effectively implement this type of instruction. This study looks at how college algebra instructors in a grant-supported professional learning community (PLC) focus on different aspects of their own and others’ teaching. We leverage the instructional triangle as an analytical framework to characterize the foci of participants’ observations. We analyzed PLC meetings where participants reported on specific aspects of each other’s observed classes. Our analysis revealed that instructors each had a primary focus that drove their observations. We anticipate these different foci will inform future PLC meetings and lead to new questions about instructor thinking, and to continued development of the instructional triangle.
In the last decade there has been a concerted effort to improve the preparation of graduate students for teaching undergraduate mathematics. Now, it is important to gauge the current state of the national landscape of graduate teaching assistant (GTA) professional development for teaching (PDT) and to assess the current needs of the community. To this end, a census survey was conducted in Fall 2021 of Ph.D-granting mathematics departments in the US. The survey followed up and extended a similar survey conducted in 2014. Some survey questions were duplicated, while others probed more deeply into the depth and extent of the GTA PDT being offered. In this paper we compare some results from the surveys, comment on how the landscape of GTA PDT has changed, and highlight areas that deserve additional attention moving forward.
It is known that teachers’ beliefs influence how they filter new knowledge and understanding of pedagogies and can directly impact their classroom practices. Moreover, the context can shape beliefs, and thus it is imperative to examine the beliefs of teaching assistants (TAs) of introductory proof courses because they play important roles in supporting student learning at this crucial mathematical junction. This preliminary study explored the professed beliefs of Lisa, a second-year, mathematics doctoral student. We qualitatively analyzed her responses in a semi-structured interview - with a particular emphasis on exploring her beliefs about teaching and beliefs about introductory proof courses. Overall, she believed that introductory proof courses pose new challenges and demands for undergraduate students. We saw connections to her beliefs about how TAs should deliver content and support students. Critical examination of TAs’ beliefs can inform how mathematics departments prepare graduate students to be teaching assistants.
Researchers typically utilize response correctness to interpret student proficiency in proof comprehension. However, metacognition has not been simultaneously analyzed alongside correctness to determine student competency in proof comprehension, yet it offers important information about performance behavior. The primary objective of this study is to investigate the accuracy of student confidence and certainty levels at local and holistic aspects of proof comprehension regarding a proof by induction. Students were given a three-factor proof comprehension assessment at the beginning and end of an undergraduate transition-to-proof course that collected student confidence, correctness, and certainty at each tier of an established proof comprehension framework. Results of this study highlight a critical distinction between high and low performers’ metacognition throughout the host course. One outlying assessment item especially illuminates additional considerations for future application of metacognition in proof comprehension research.
Calculus courses are an integral component for STEM degree attainment yet are historically attributed to high attrition rates and lowering student confidence. Although prior research identified coordination of instruction as a successful characteristic of Calculus programs, there is a need to further examine the role of Instructor Autonomy (e.g., decision making on content and instruction) and its potential impact on student perceptions of the climate. We analyzed data from two national studies of introductory math courses focusing on instructor (N=321) reports of course decision making and student (N=14,488) reports of climate. Preliminary results indicated a great deal of variability in Instructor Autonomy across the Precalculus to Calculus 2 sequence within our sample and even within a department. Narrowing our focus to Calculus 1, variability in Instructor Autonomy persisted and student perception of climate was not unified; however, courses that utilized team-based decision making promoted higher student perceptions of climate.
Assessing student understanding of calculus concepts is of interest to RUME researchers and practitioners alike. This paper details the initiation of the development phase of a new instrument to assess student understanding of the central concepts in the introductory collegiate calculus course. We discuss a framework for defining “conceptual understanding” and make transparent our process for selecting and refining robust tasks to measure it. We provide discussion on interview responses that reveal how students may attempt to rely on memorized procedure and, as a result, discredit productive conclusions. Finally, we outline and seek discussion on future work to continue development of valid and reliable questions to assess the broad range of productive understandings in calculus.
There is currently little research that examines the interplay between various affective constructs, particularly within the context of mathematics assessment. This longitudinal study seeks to understand how student assessment-related affect changes across a university mathematics course and seeks to identify latent or explicit factors that play a role. We collected data on achievement emotions, self-efficacy, stress, and stress mindset across three time points, in addition to measures of academic performance, specifically on different forms of assessment. Without accounting for interactions, preliminary results considering the constructs individually demonstrate an increase in negative and decrease in positive affect. However, incorporating path analysis suggests that engagement on low-risk summative assessment has the potential to promote self-efficacy. As a consequence of this research, we hope to pinpoint suitable interventions that may have a positive influence on affect and achievement. The disentangling of these relationships will have practical implications for course design and assessment structure.
Calculus remains a major barrier of entry into STEM for students without strong mathematics backgrounds. This study looks at one implementation of the STEM Core model, which provides an accelerated pathway through developmental math classes along with extensive support to get students calculus ready in one year. Viewing this program as a community of practice around math learning I examined course completion data and conducted interviews to understand how the model is impacting student success. Furthermore, I seek to understand what aspects of the model matter most to the students. Initial results of student success rates in the course sequence show significant positive outcomes, with nearly 65% of STEM Core students reaching Calculus I in one year. Interviews with students reveal how their success goes beyond course completion and the importance of the relationships they built with their instructors and one another.
We address the transformation of instruction in entry-level college mathematics courses, such as College Algebra. Our research question is: What do instructors view as benefits and challenges when implementing novel digital graphing activities in College Algebra? We report on a case study of two instructors who implemented the activities during both semesters of one academic year, drawing on instructors’ individual interviews at the end of each semester. The instructors viewed it as beneficial to implement these activities as part of a community. They also found the activities’ focus on reasoning helpful for their students. They found integrating the new activities with existing online learning management systems challenging at times, and they wished their students were more engaged when implementing activities asynchronously. Overall, the challenges were not roadblocks, and the benefits outweighed the challenges. We conclude with discussion and implications for research and practice.
This work is part of a broader project to investigate student understanding of mathematical ideas used in upper-division physics. This study in particular probes students’ understanding of the divergence and curl operators as applied to vector field diagrams. We examined how students reason with partial derivatives that constitute divergence and curl of the vector field diagrams. Students’ written responses to a task on derivatives, divergence, and curl of a 2D vector field were collected and coded. Students were generally successful in determining the sign of some of the constituent derivatives of div and curl, but struggled in one case in which components were negative. Analysis of written explanations showed confusion between the sign, direction, and change in the magnitude of vector field components.
One of the primary mathematical practices of many professional research mathematicians is engaging with highly advanced mathematical concepts in their research. As mathematicians engage with these concepts in their work, they may or may not ‘fully understand’ them, which could include not being able to deduce why each step in a proof logically follows from the next. This preliminary study investigates what reasons professional mathematicians give for not ‘fully understanding’ mathematical concepts in their own work. One mathematician researching coding theory was interviewed, which utilized one of the participant’s own research journal articles as a launching point for discussion. Various reasons for the participant’s conceived amount of understanding were observed, including authoritative reasoning and academic pressures. Implications for future research on professional mathematicians are discussed.
We present preliminary results of students’ strategies playing Vector Unknown: Echelon Seas [VUES], a 3D videogame intended to support student reasoning about vectors. Our team designed VUES by drawing on theories from Inquiry-Oriented Instruction (IOI), Game-Based Learning [GBL] and Realistic Mathematics Education [RME]. VUES builds from a prior 2D game by giving players vectors with 1, 2, or 3 components, depending on the level. We use codes from our team’s prior analysis (Mauntel et al, 2020) to analyze strategies in the 3D game. Early results show that students develop similar strategies during 3D gameplay as other students developed while playing the 2D game. However, we have also found new strategies that we did not witness with 2D gameplay, requiring us to extend our coding scheme. Further, early results emphasized the need for design changes to the 3D game to better support players’ progress.
Dismantling notions of objectivity in the qualitative research process necessitates engaging research subjectivity in meaningful and explicit ways. Poetic transcription methodology allows for an entry point into conversations around research subjectivity. In the context of a critical exploration into students’ definitions of their own mathematical success, the poetic transcription results of two researchers are comparatively analyzed to highlight structural and thematic (mis)alignments in researchers’ poetic interpretations of students’ definitions of mathematical success. We emphasize both the scholarly value and uncertainty that the results of such a comparison produce and discuss how future work might build on these comparisons and on our understanding of what poetic transcription can bring to mathematics education.
This study explored pre-service teachers’ (PSTs’) initial experience connecting mathematics and social justice. This work is part of an on-going action research project on improving mathematics PSTs’ experiences in their methods courses, specifically focusing on culturally responsive mathematics teaching. Research stresses the need to prepare teachers who can teach a diverse student body, are mindful of differences in student backgrounds and, are cognizant of their own biases (Mark & Id-Deen, 2020; Brown et al., 2019; Gallard, Mensah, & Pitts, 2014; Johnson & Atwater, 2014; Lemons-Smith, 2013; Leonard, 2008; Lewis, Pitt, & Collins, 2002). PSTs developed lesson plans balancing both mathematics and social justice goals. Lesson plans were analyzed to find initial connections PSTs made between mathematics and social justice.
Most upper-division physics courses use mathematics beyond that encountered in the calculus sequence. Matrix multiplication and eigentheory are used by physics students in courses including classical mechanics, optics, and quantum mechanics. While there have been recent investigations of student use of linear algebra in quantum mechanics, no prior work has examined how physics students approach this mathematics in other contexts. In this paper, we investigated the extent to which student interpretations of matrix multiplication could be characterized by those from Larson and Zandieh (2013): linear combination, system of equations, and transformations. We have examined student responses to written questions and used the results to develop an interview protocol to examine how physics students interpret matrix equations using the three interpretations to classify their responses. Results are shown from an interview sample of six undergraduate physics majors.
Calculus courses should embrace innovative pedagogies that promote Latin* students’ mathematics identity development. One such pedagogy is specifications grading, which might foster students’ growth-mindset and perseverance in mathematics. In this study, we examine the mathematics identity development of two Latin* students’ in Calculus courses with specifications grading, and we compare this development to that of a student not enrolled in a specifications grading course. We compare the changes in the students’ mathematical competence and the changes in their sense of recognition as a doer of mathematics. We discuss how aspects of the specifications grading pedagogy can contribute to students’ mathematics identity development.
Small-scale teaching experiments focused on how students reason about exponential functions provide an incredibly useful foundation when thinking about the design of classroom tasks. However, scaling-up certain critical features of these experiments such as focused and prolonged instruction on a single topic is unrealistic for those teaching at the university level. Drawing on research on covariational reasoning, exponential functions, and design-based research methodology, we developed and implemented two interactive lessons with the goal of enhancing students’ understanding of exponentiation and covariational reasoning. In particular, we sought to identify the quantities and relationships that students focused on during classroom tasks and the ways in which the tasks could be refined to support students’ understanding of exponential covariation. From this analysis, revisions made to the curricular materials and other implications for research and teaching are discussed.
The relationship between active learning and equity is a prominent discussion within mathematics education, as it should be. There are conflicting findings on whether active learning promotes or detracts from equitable outcomes. This study aims to highlight student voices in describing the student experience in active learning advanced mathematics classrooms. This preliminary report outlines key themes that surfaced from a cursory analysis of the data; quiet students were seen as less confident or less competent by their peers during whole class discussions, and students were keenly aware of the knowledge distribution within small group discussions. It is important to understand the implications of the sociocultural dynamics that may be emphasized in active learning classrooms, thus being better equipped to attend to equity.
Mathematics educators assign important learning outcomes to abstract algebra classes. However, it is unclear whether students share these goals or whether they have different motivations entirely. Using the framework of expectancy-value theory, we highlight preliminary results from semi-structured entrance and exit interviews with six undergraduates enrolled in abstract algebra. This report centers around the following research question: What are students’ expectations, motivations, and goals for their undergraduate abstract algebra course?
Meta-Representational Competence (MRC), first theorized by Andrea diSessa, has been used widely in Math Education studies. It has been recently used in Physics Education Research in the domain of Quantum Mechanics. Specific problems within Quantum Mechanics have a foundation in Linear Algebra, and may be approached or perceived differently based on the notation used (either Dirac, Matrix, or Spinor notation). Semi-structured interviews were conducted to present physics students with content questions and then ask them about MRC concepts directly, as it related to the content questions or other situations in Physics or Math. Student statements were coded to indicate both previously identified and novel facets of MRC in the context of change of basis problems in Quantum Mechanics. The preliminary analysis of two student interviews demonstrates each student’s distinct utility of MRC concepts, and suggests extending the Wawro, Watson, & Christensen (2020) MRC statement codes.
The growing call for more equitable and just mathematics classrooms includes supporting and strengthening students’ mathematical identity. Recent scholarship proposes a connection between students’ mathematical identities and their values (Kalogeropoulos & Clarkson, 2019). However, limited research acknowledges and explores mathematics identity at the intersection of gender and race (Leyva, 2021; McGee, 2016). Furthermore, undergraduate mathematics in the United States often does not reflect the cultural values of women or students of color (Fong et al., 2019; Leyva, 2016, 2021). Informed by sociopolitical theory and an intersection lens, this study explores women of color’s mathematics education values and the connections between those values and their mathematical identity. Through interviews and mathematical autobiographies, participant examples indicate various values, including those related to learning from mistakes and social support. Preliminary results suggest that the presence of these values in mathematics environments may support women of color’s mathematical identity.
As English is increasingly seen as the language of mathematics instruction in the United States and the world, it becomes important to interrogate the role of the English language in mathematical understanding. This article investigates this question in relation to linear (in)dependence and limit through clinical interviews with undergraduate students. Preliminary results indicated a relationship between students’ conceptual understanding of linear (in)dependence and their ability to connect their mathematical and non-mathematical uses of the term (in)dependent. Moreover, students’ unproductive conceptions of limit appeared to be influenced by non-mathematical meanings of limit.
I present preliminary results from my dissertation research that investigates how eight college Calculus I instructors used representations in an instructional task intended for introducing derivatives with inquiry. Using Zandieh’s (2000) framework, the instructors were asked to propose and sequence eight tasks for introducing derivatives to their students. This paper focuses on the instructors’ utilization of various representations of derivatives in the first task they proposed in the sequence. The findings show how all instructors found graphical and physical representations appropriate for introducing derivatives, with some of them also reaching out to other representations (symbolic, numerical, and verbal) simultaneously.
Postsecondary instructors interested in inquiry-oriented instruction of Linear Algebra participated in a sequence of eight one-hour online work group meetings with other experienced inquiry-oriented linear algebra facilitators and teachers. Recordings from three meetings were analyzed for how two participants referenced goals of instruction in preparation for teaching a new instructional unit on subspaces. We identified four goals of instruction of teaching subspaces. We discuss the intersections of several goals of instruction and possible implications for those who want to transition to inquiry oriented instructional approaches.
Conceptualizing and expressing distances on graphs of functions are central to understanding why integrals afford measuring areas and volumes in calculus. In this preliminary report, we share the results of implementing an intervention designed to support these skills with 31 Integral Calculus students. Initial results show that the activity did support some students in learning to accurately expressing the horizontal distance between two functions. However, a closer look at students’ work revealed that the tasks may not have supported all of the ways of reasoning we intended. We discuss the instance of two students who successfully completed the activity yet displayed evidence of treating symbols as labels or parameters and following a pattern to complete the tasks rather than conceptualizing distance. We close with a discussion of how to improve the next iteration of the intervention.
We share preliminary results of a commognitive analysis of the participation of twelve undergraduate students in an exploratory subgroup generation activity in an abstract algebra class. We find that while students’ collaborative activity contains markers of both ritualized and explorative participation, their appeals to ritual and personalized formulations of inferences and insights about subgroups serve to broaden access to the conceptual substance of the activity. We discuss a possible asset-oriented interpretation of students’ ritualized activity as an aid to exploration and reflective of the disciplinary activity of professional mathematicians.
We report on work to explore instructor intercultural competence and its relationship with the effectiveness of mathematics teaching in post-secondary settings. Results are from quantitative and qualitative inquiry into the mathematical knowledge for teaching and intercultural orientations of 15 instructors and 476 of their undergraduate students. Instructors participated to varying degrees in a professional short-course for faculty learning to teach mathematics for future grade school teachers. The greatest learning gains among undergraduates were in the classes of those instructor-participants who completed short-course activities about interculturally responsive mathematical knowledge for teaching future teachers.
The work of Imre Lakatos has been influential in the philosophy of science and mathematics. His writings on how science and mathematics progress as fields have been utilized in mathematics and science education. In this study, we apply Lakatosian theory in analyzing task-based interviews. We use conflict diagrams to center the discrepant events participants experience while solving calculus tasks. In so doing, we illustrate how problem context plays a vital role in the evolution of problem-solver's structures in problem situations.
This report describes the preliminary results of a genre analysis of mathematical remarks, which was conducted using Bhatia’s (1993) seven-step genre analysis model. As part of the analysis, discussion threads on Mathematics Stack Exchange about mathematical remarks were analyzed as well as a corpus of remarks from first-year mathematics lecture notes of a British university. An overarching communicative purpose and seven non-mutually exclusive “mini communicative purposes” (Hyon, 2018, p. 30) of the genre of mathematical remarks are identified
The Algebra Instruction at Community Colleges: Validating Measures of Quality Instruction project (VMQI) developed an instrument to measure mathematical knowledge for teaching community college algebra (MKT-CCA) and conducted cognitive interviews with 12 community college instructors of College Algebra. Thirty-six drafted MKT-CCA test items were reviewed with each item being reviewed by two instructors. This paper presents the lessons learned from this preliminary analysis of the cognitive interview transcripts.
This study explores a mathematics department’s advising process for incoming, first-year undergraduate students. Through the analysis of video-recorded advising sessions, the interactional dynamics of an undergraduate mathematics advising program are explored. In particular, we describe how student-advisor interactions co-construct students’ course-taking narratives in ways that shape their mathematical opportunities.
Communities of practice (CoPs) offer a structure for individuals with common goals to engage in collective learning (Wenger-Trayner & Wenger-Trayner, 2015). The CoMMIT Network was developed to support regional Math CoPs of undergraduate mathematics faculty interested in using active learning and inquiry teaching approaches in their courses. This study utilized the value framework (Wenger et al., (2011) to interrogate the value participating in the CoPs and COMMIT Network provides participants.
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Existing research indicates that some learning challenges Calculus students face may not be related to the Calculus-specific content, but rather from issues related to prerequisite knowledge. Several Calculus topics such as volume by slicing and volumes of revolution have a prerequisite topic of volume. These Calculus topics tend to be challenging for students in various ways. Better understanding how these students describe and use volume may help inform instructors as well as potential reforms to practices to better support student learning of these Calculus topics.
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Contributing to the growing body of research on opportunities to learn provided by undergraduate mathematics textbooks, this study reports on the analysis of optimization examples given in a commonly used business calculus textbook in the United States. Findings of this analysis indicate that the textbook consistently encourages students to interpret critical numbers and extrema, in addition to encouraging students to verify whether or not critical numbers produced minimum/maximum values of the objective functions from which they were determined. The analysis further revealed that the textbook rarely encourages students to include units of extrema, provides limited opportunities for students to reason about objective functions that have more than one critical number, and provides minimal opportunities for students to reason about absolute extrema problems, respectively. Recommendations for different stakeholders, including business calculus textbook authors and business calculus instructors, are included.
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The results presented here offer preliminary insights into the course patterns and academic achievement of students taking entry-level math courses at Johns Hopkins University (JHU) from Fall 2017 through Fall 2021. First, we used students’ math placement exam results and end-of- semester grades in their first math course to predict academic achievement when considering whether students followed their placement recommendation. Second, we compared trends in placement data and academic achievement of students taking math courses at JHU before and during the COVID-19 pandemic.
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Assessing students’ metacognition can help educators better understand student learning. In the present study, the authors evaluated students’ pre-assessment confidence, response correctness, and post-response certainty during routine examinations in an undergraduate linear algebra course to investigate the magnitude and direction of students’ metacognitive shifts ( meta) from mean confidence to mean certainty as compared to response correctness. Results of this study provide insights to student perceptions of their knowledge during examination performances in linear algebra and can inform efforts to improve mathematics teaching and learning.
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