Undergraduate Calculus Research Articles

Teaching dynamics to enhance critical thinking and knowledge socialization in the mathematics classroom

Building the future of education together means that, as educators, we need to promote innovation and educational strategies in our classroom to foster the development of skills such as critical thinking and argumentation. Moreover, we should encourage using technology as an efficient and effective tool to solve complex problems. Mathematics is a cornerstone in the education of engineering students. However, students’ mathematic classroom experiences have only presented them with a rigid and pure side to the discipline. Active learning strategies foster the development of competencies such as critical thinking and argumentation while generating knowledge socialization and learning democratization. The active learning dynamics were conducted in first-year undergraduate calculus course for engineering and are based on collaborative learning, model-eliciting activities, and the use of technology. Students reported that being actively involved and interacting with classmates and tools made them enjoy the class, broaden their view of mathematics, and connect with contexts from other disciplines. Moreover, these dynamics favor different types of formative assessment for students, self- and peer-assessment, and for teachers about the students learning and the design of the activity. Based on educational research, the design of the dynamics is innovative in the combination of implementations used to potentiate the benefits of a student-centered learning environment and the socialization of knowledge.

A Flipped Class to Support the Success of At-Risk Students

Early identification of at-risk students for timely intervention is critical to prevent non-completion of study programs. This article proposes a flipped class framework to support the academic success of at-risk students in an undergraduate Calculus course. It comprises three main components of setting, conduct, and monitoring. A flipped support class was implemented as periodic sessions throughout the learning semester over six consecutive semesters for the selected total of 560 at-risk students. At-risk students who attended the flipped support class reported a higher passing percentage than those who did not, in each of the six semesters. A strong mathematics foundation contributed to the likelihood of passing the course; however, it can be further increased by attending more hours of the flipped support class.

Unveiling interdisciplinary horizons: students’ experiences in a first-year calculus course

In the realm of higher education, the pursuit of interdisciplinarity aims to foster the exchange and integration of fragmented knowledge, yielding transformative outcomes. Employing a phenomenological method, this study delves into the interdisciplinary experience of twelve students from a first-year undergraduate calculus class. Through the analysis of written questionnaires, focus group sessions, and supplementary qualitative data, a comprehensive understanding of students’ interdisciplinary encounters is unveiled and organized into three main categories: what students think about interdisciplinarity, how they act when being involved in integrations and what external factors are involved in shaping their experience. This paper presents emergent experiential themes, shedding light on both individual and collective experiences, as students navigate and enrich their calculus learning through interdisciplinary connections.

An intuitive and simplified formulation of general relativistic gravity in freely falling frames

General relativistic gravity in freely falling frames is formulated using undergraduate calculus and matrix algebra. The key to the formulation involves retaining Newton’s first law of motion in the local spacetime of freely falling frames. The formulation shows that the relative acceleration of two particles in the freely falling frame does not depend directly on their relative velocity. Also, the 4 × 4 matrix relating the relative acceleration of two particles to their relative spacetime displacement closely resembles the physics of Newtonian gravity, implying that the local gravitational field obeys Laplace’s equation. The results of this paper and a previous publication [L. R. Miller, Phys. Essays 35, 202 (2022)] provide an intuitive approach that enables the use of lower-level mathematics to arrive at Einstein’s theory of gravity in empty space.

Opportunity to learn about optimization problems provided by undergraduate calculus textbooks: A case study

This study investigated opportunities to learn about optimization problems provided by undergraduate calculus textbooks. To accomplish this, we analyzed examples and practice problems from two calculus textbooks widely used in the teaching of calculus in the United States. Findings of this study indicate that a majority of the problems in both textbooks lack realistic and essential contexts, have matching or missing information, and require a low cognitive demand to solve. Setting up objective functions is either not necessary or it is straightforward for most problems in the two textbooks. In addition, both textbooks provide ample opportunities to interpret critical numbers and extrema in context. Overall, findings of this study suggest the existence of a relationship between known students’ difficulties with solving optimization problems and the opportunities to learn about optimization problems provided by calculus textbooks. Implications for several stakeholders, including calculus textbook authors and calculus instructors are discussed.

Women’s Sense of Belonging in Undergraduate Calculus and the Influence of (Inter)Active Learning Opportunities

Women continue to be underrepresented in undergraduate STEM majors.<b> </b>Prior studies identify sense of belonging, or the extent to which one feels like an accepted member of an academic community, as a key contributor to women’s decisions to stick with or leave their STEM majors. Calculus can be an especially critical leak in the STEM pipeline, as is often seen as a gatekeeper course for STEM majors. Historically, Calculus instruction has been primarily lecture-based, though recently, efforts have shifted toward incorporating instruction that supports active learning. Prior studies have suggested that providing active learning opportunities may support students’ sense of belonging. However, there lacks consensus on the particular types of active learning opportunities that best support students. This mixed methods study investigates connections between women’s sense of belonging and the learning opportunities they experience during the first semester of a two-semester Calculus course designed to provide frequent active learning opportunities. Findings indicate a significant increase in sense of belonging, perceived competence, and social connectedness from Week 1 to Week 7 of the semester. Further, women overwhelmingly identified the interactive nature of group work and interactive lecture as supportive of their sense of belonging for both academic and social reasons.

On the bifurcation behavior of a folded notebook page

We study how a bound notebook page can be folded once so that the page is visible when the notebook is closed and becomes a bookmark. An additional practical requirement is that the folded page stays within the same side of the bind, so that it does not get creased further when one closes the notebook. This simple problem displays a remarkably rich bifurcation behavior, which can be analyzed using undergraduate calculus. The main finding is that the optimal way of folding the page depends on the page's aspect ratio and changes abruptly when the latter exceeds the critical value of 1.207 11. This behavior also resembles a first-order phase transition in thermodynamics despite its geometric origin.

Motivation and achievement in undergraduate STEM calculus: Applying self‐determination theory to examine the differential impact of learning environments across demographic groups

AbstractThis study examines the relationships among students' perceptions of autonomy, competence, and relatedness, self‐determined motivation, and achievement across two different undergraduate Calculus learning environments at the same institution. Furthermore, it also exposits how learning environments are not culturally or socially neutral by establishing clear evidence that one learning environment offered motivational support and associated achievement gains to specific demographic groups while diminishing motivation and achievement for others. To accomplish this, we applied constructs from Self‐Determination Theory (SDT) within a quasi‐experimental design (N = 6866), controlling for prior math knowledge, in both Calculus I and Calculus II over seven semesters. We use this information to advance the broad notion of student‐centeredness and identify the role of motivational processes by which undergraduate Calculus learning environments function. We propose using motivational frameworks as a tool for defining and measuring student‐centeredness relative to individual students, or student groups, rather than to students as a monolith.

Teaching Undergraduate Calculus at an Urban HBCU through a Global Pandemic

Over the last decade, there has been a national effort to prepare a more diverse student population for success in science, technology, engineering, and mathematics (STEM) careers. While there are a variety of best teaching practices in undergraduate STEM education, student engagement has been widely used to improve student learning. Specifically, it is important for students to have opportunities to practice or apply the theory they have learned in courses to real-world problems. Implementing active learning is one model that can be used to achieve this objective. However, due to the COVID-19 pandemic, professors have had to offer courses in online formats and determine how to continue to implement effective teaching strategies during this unprecedented time. Professors are anticipating the possibility of the need to continue offering courses in fully online and hybrid formats even in the future. With an emphasis on a STEM gateway course sequence, the aim of this paper is to highlight pre-COVID-19 research studies in online STEM education at Historically Black Colleges and Universities (HBCUs), explore how COVID-19 impacted students, teaching, and learning in undergraduate calculus courses, and outline opportunities for continuing to implement teaching techniques that work effectively for in-person and online calculus instruction.

Examining Students’ Concept Images in Mathematics: The Case of Undergraduate Calculus

Examining Students’ Concept Images in Mathematics: The Case of Undergraduate Calculus

The role of leadership in educational innovation: a comparison of two mathematics departments’ initiation, implementation, and sustainment of active learning

Several studies have shown that the use of active learning strategies can help improve student success and persistence in STEM-related fields. Despite this, widespread adoption of active learning strategies is not yet a reality as institutional change can be difficult to enact. Accordingly, it is important to understand how departments in institutions of higher education can initiate and sustain meaningful change. We use interview data collected from two institutions to examine how leaders at two universities contributed to the initiation, implementation, and sustainability of active learning in undergraduate calculus and precalculus courses. At each institution, we spoke to 27 stakeholders involved in changes (including administrators, department chairs, course coordinators, instructors, and students). Our results show that the success of these changes rested on the ability of leaders to stimulate significant cultural shifts within the mathematics department. We use communities of transformation theory and the four-frame model of organization change in STEM departments in order to better understand how leaders enabled such cultural shifts. Our study highlights actions leaders may take to support efforts at improving education by normalizing the use of active learning strategies and provides potential reasons for the efficacy of such actions. These results underscore the importance of establishing flexible, distributed leadership models that attend to the cultural and operational norms of a department. Such results may inform leaders at other institutions looking to improve education in their STEM departments.

Observing Instructor Behaviour in an Active Learning Classroom: A Case Study of an Undergraduate Calculus Course

Active learning classrooms (ALCs) are spaces explicitly designed to encourage collaborative learning, often through the use of technology. To learn more about teaching activity in ALCs, a study was designed to observe an engineering calculus course during the winter 2020 term. A large-scale active learning classroom was selected for classroom observation using the extended Teaching Dimensions Observation Tool (TDOP+). The TDOP+ is a descriptive classroom observation protocol based on the Teaching Dimensions Observation Tool, enhanced with elements from the Active Learning Classroom Observation Tool (ALCOT). 
 This case study compares the class orchestration of different instructors teaching two different sections of the same course at a large, public university. Twenty class sessions were coded for this study: 10 for each section (5 for each instructor). The coded instructor behaviour was analyzed using a conceptual framework described by Nocera (i.e., a version of Activity Theory), focusing on mediating artifacts and instructor goals. 
 While we observed differences in the frequency and duration of active learning activities and in the type and number of tools used in each class session, the results from this case study suggest that flexible space design enables instructors with the same lesson plan (and content) to create different technological frames to achieve their varied pedagogical goals, while encouraging increased adoption of new tools.

A blind spot in undergraduate mathematics: The circular definition of the length of the circle, and how it can be turned into an enlightening example

We highlight the fact that in undergraduate calculus, the number pi is defined via the length of the circle, the length of the circle is defined as a certain value of an inverse trigonometric function, and this value is defined via pi, thus forming a circular definition. We present a way in which this error can be rectified. We explain that this error is instructive and can be used as an enlightening topic for discussing different approaches to mathematics with undergraduate students.

An alternative way of defining integration in multivariable calculus

In undergraduate calculus of several variables, double and triple integrals are usually defined as limits of certain Riemann sums. The existence of the integral, as well as the integration formulas, are stated without proof since they require more advanced mathematics. In this article, an alternative and straightforward way of defining multiple integrals is proposed where the usual integration formulas follow as a direct application of the definition. The underlying idea is to map the arbitrary region of integration to an n-dimensional open interval, and integration over the latter is defined via the usual iterated integral. Moreover, the substitution formula is taken as a definition. Numerous illustrative examples are provided.

An intuitive and simplified formulation of general relativistic gravity in empty space

General relativistic gravity in empty space is formulated using undergraduate calculus, differential equations, and matrix algebra. The key to the formulation involves certain limited assumptions of Newtonian-like behavior in freely falling frames. The formulation results in a field-motion matrix equation that serves essentially the same purpose as Einstein’s field equations in empty space. The limited assumptions of Newtonian-like behavior in freely falling frames are then justified using well-known tensor relationships. To demonstrate its use, the new field-motion equation is used to solve for the Schwarzschild metric.

“They’re just students. There’s no clear distinction”: a critical discourse analysis of color-evasive, gender-neutral faculty discourses in undergraduate calculus instruction

ABSTRACT Background Calculus instruction is underexamined as a source of racialized and gendered inequity in higher education, despite research that documents minoritized students’ marginalizing experiences in undergraduate mathematics classes. This study fills this research gap by investigating mathematics faculty’s perceptions of the significance of race and gender to calculus instruction at a large, public, historically white research university. Methods Theories of colorblind racism and dysconsciousness guided a critical discourse analysis of seven undergraduate calculus faculty’s perceptions of instructional events. Findings Our analysis revealed two dominant discourses: (i) Race and gender are insignificant social markers in undergraduate calculus; and (ii) Instructional events can be objectively deemed race- and gender-neutral. We illustrate how calculus faculty varyingly engaged these colorblind discourses as well as discourses that challenged such conceptions of instruction. We also highlight how faculty dysconsciousness in reports of instructional practices reflect potential operationalization of dominant discourses that reinforce colorblind racism. Contribution With limited research on faculty perspectives on racial equity in mathematics, our study documents how color-evasive, gender-neutral discourses among mathematics faculty shape orientations to instruction that reinforce the gatekeeping role of calculus in STEM higher education. Implications are provided for race- and gender-conscious undergraduate mathematics instruction and faculty development.

Support for All? Confronting Racism and Patriarchy to Promote Equitable Learning Opportunities through Undergraduate Calculus Instruction

Support for All? Confronting Racism and Patriarchy to Promote Equitable Learning Opportunities through Undergraduate Calculus Instruction

You Don’t Want to Come Into a Broken System: Perspectives for Increasing Diversity in STEM Among Undergraduate Calculus Program Stakeholders

You Don’t Want to Come Into a Broken System: Perspectives for Increasing Diversity in STEM Among Undergraduate Calculus Program Stakeholders

The impact of prerequisites for undergraduate calculus I performance

We conducted a quantitative analysis to determine how the prerequisite path of students taking calculus I impacts their grade performance. We began by investigating the performance of students that took college algebra and trigonometry versus those that took pre-calculus ahead of their credit-bearing calculus I attempt. We concluded that there was a significant difference between the two prerequisite routes. We then performed regression analysis to view the number of credit prerequisite credit hours, including multiple attempts, as a predictor of calculus I GPA and A-proportion. We found a strong negative correlation between these variables. We hope this study can be replicated at other institutions and in other fields to help university policymakers with decision-making regarding course listings.

How to correctly answer ‘Is the optimal critical point a local minimizer?’ in Calculus courses

This short note discusses how the optimality conditions for minimizing a multivariate function subject to equality constraints have been covered in some undergraduate Calculus courses. In particular, we will focus on the most common optimization problems in Calculus of several variables: the 2 and 3-dimensional cases. So, along with sufficient conditions for a critical point to be a local minimizer, we also present and discuss counterexamples for some statements that can be found in the literature of undergraduate Calculus related to Lagrange Multipliers, such as ‘between the critical points, the ones which have the smallest image (under the function) are minimizers’ or ‘a single critical point (which is a local minimizer) is a global minimizer’.