Mathematics Commons^™

A Modeling Scenario For Cooling A Hot Vehicle In Florida, Jared Bunn, Bernadette Mullins, Elizabeth Hale, Jaeyoun Oh

CODEE Journal

This paper presents a group project assigned in a Calculus 2 course that has students work to develop, analyze, and draw conclusions about a modeling scenario for cooling a hot car. Using a modeling-first approach, instructors supported the students in class throughout the beginning of the project, enabling the groups to complete the remainder of the project on their own. Students used parameter estimation to tune their models to provided data: one for windows being up, and one for windows being down. This project provides an example of how modeling can be introduced early in a calculus course, rather than …

Math 141 Calculus I (Queens College), Anisha Clarke

Open Educational Resources

This syllabus provides the outline for a calculus one course using OpenStax Calculus Volume I and no graphing calculator.

A Two-Phase Perspective On A Related-Rates Paradox, Charlie Vazquez Acosta

Honors Capstones

Related-rates problems are a standard topic in first-year calculus and have appeared in textbooks for over 150 years. These problems are used to teach implicit differentiation and the relationship between changing quantities. Common examples include the falling ladder, the fishing bobber, the melting snowball, and the leaking conical tank. In each of these problems, a quantity is changing at a constant rate, and students are asked to find the rate of change of another related quantity. While the computations themselves are usually straightforward, the standard models lead to unrealistic results near the end of the motion. For example, the falling …

Instructors' Lived Experiences Of Using Authentic Assessments In College Calculus Courses Across The United States: A Phenomenological Study, Christy Hart Kains

Doctoral Dissertations and Projects

The purpose of this transcendental phenomenological study was to describe the lived experiences of instructors with authentic assessments in college calculus courses across the United States. The theory guiding this study was Astin’s student involvement theory, which provided a framework to study the involvement of students in higher education. The central research question for this study was: What are the lived experiences of instructors who utilize authentic assessments in college calculus courses across the United States? This study used a qualitative design following the transcendental phenomenological approach of Moustakas. The sample included 12 instructors from two-year and four-year colleges in …

Designing An Approach To Developing Students' Symbol Sense For The Derivative In Calculus, Laura E. Weinstein

Theses, Dissertations and Culminating Projects

Students often struggle to connect the conceptual meaning of the derivative with the symbolic structures of Leibniz (dy/dx) notation. Although symbol sense has been described as the coordination of mathematical concepts, signs, and objects, little is known about how instruction can purposefully support students in developing symbol sense for the derivative. This study addresses this gap by designing and examining a learning approach aimed at helping students construct the relational structures underlying derivative notation. Using design research methodology, the study engaged intact algebra, precalculus and calculus classes in a suburban New Jersey high school across iterative cycles of design, implementation, …

Student Conceptions Of Summation And Limits In The Definite Integral Following Quantitatively Focused Instruction, Caleb Daniel Holloway

2026 Scholarly Teaching Conference: Concurrent Session Papers

Recent studies have examined how students form productive conceptions of the definite integral and discussed techniques for promoting such conceptions. In this paper I present findings from interviews held with six students enrolled in second-semester calculus, four of whom had received quantitatively focused instruction on the definite integral. All six were chosen for their observed use of summation conceptions on definite integral problems, and here their conceptions are explored further. Additionally, we gain insight on their thinking regarding limits as related to the definite integral. The findings presented here add to our understanding of student thinking regarding the integral and …

Active Calculus: Single Variable, 2nd Edition, Matthew Boelkins, David Austin, Christina Safranski, Steven Schlicker

Open Textbooks

Active Calculus is different from most existing calculus texts in at least the following ways: the text is freely readable online in HTML format and is also freely available for in PDF; in the electronic formats, graphics are in full color and there are live links to java applets; there are live WeBWorK exercises in each chapter, which are fully interactive in the HTML format and included in print in the PDF; the text is open source, and interested users can gain access to the original source files on GitHub; the style of the text requires students to be active …

Shaping Mathematics Identity: An Exploratory Study On Specifications Grading In Calculus I At A Hispanic-Serving Institution, Luis M. Fernandez, Kaitlyn Stephens Serbin, Cristina Villalobos, Shaghayegh Azadi Setayesh, Guillermo Garza

School of Mathematical & Statistical Sciences Faculty Publications

Calculus I courses play a pivotal role in shaping students' STEM pathways, making it essential to adopt pedagogies that foster both achievement and mathematics identity development, particularly among underserved groups such as Hispanic students. This study explores the impact of Specifications Grading, an alternative assessment method where students meet specific course learning objectives through multiple attempts, on students’ mathematics identity development. Through a comparative case study of two Calculus I students at a Hispanic-Serving Institution, one enrolled in a specification graded course and the other in a traditionally-graded course, we examine shifts in their self-perceptions of competence, interest, recognition in …

Oer Textbook Review For Calculus - Openstax Calculus, Jing Hu Ph.D.

Open Educational Resources Publications

This OER textbook review provides a comprehensive evaluation of the "Calculus" textbook series published by OpenStax. The reviewer, Jing Hu, an adjunct lecturer at Bentley University, highlights the textbook's strengths, including its thorough coverage of essential calculus topics, accurate and well-established mathematical principles, practical relevance, and user-friendly design. The open-access nature of the resource is seen as a significant advantage, contributing to its long-term utility and accessibility for both students and educators. Overall, the review concludes that the OpenStax Calculus textbook is a high-quality, comprehensive, and freely available resource that effectively supports the learning and teaching of calculus.

Calculus Students’ Problem-Solving Strategies On Related Rates Of Change Problems Appearing In Online Versus Paper-And-Pencil Format, Tyson Bailey

Mathematics Dissertations - Archive

This study explores first-semester calculus students’ use of mathematical problem-solving strategies while working related rates of change problems in both an online homework format and a traditional pencil-paper format. We address two research questions: (1) How do students’ mathematical problem-solving strategies when working online homework on related rates of change problems compare with their problem-solving strategies when working paper-and-pencil homework related rates of change problems? (2) What influence does the ‘view an example’ feature in online homework have on a student’s problem-solving strategies when working an online RRC homework problem? Using scores on free-response midterm exam problems on related rates …

Differential Calculus: From Practice To Theory, Eugene Boman, Robert Rogers

Milne Open Textbooks

Differential Calculus: From Practice to Theory covers all of the topics in a typical first course in differential calculus. Initially it focuses on using calculus as a problem solving tool (in conjunction with analytic geometry and trigonometry) by exploiting an informal understanding of differentials (infinitesimals). As much as possible large, interesting, and important historical problems (the motion of falling bodies and trajectories, the shape of hanging chains, the Witch of Agnesi) are used to develop key ideas. Only after skill with the computational tools of calculus has been developed is the question of rigor seriously broached. At that point, the …

Wicked Problems And The Invention Of Calculus, Ernesto Diaz

Natural Sciences and Mathematics | Faculty Scholarship

Since the 1980s, wicked problems have represented a category of challenges that defy clear description, cannot be addressed with existing models or theories, and resist experimentation in trying to solve them. This class of problems existed before they were identified and have been unsuccessfully addressed with Thomas Kuhn’s model of scientific discovery, an expectation that requires the identification of a new object and the development of its correct interpretation. This paper proposes an alternative view of scientific discovery using the invention of Calculus as a case study that describes a successful process addressing wicked-like problems from a philosophical perspective, develops …

Broadening Views Of Mathematical Creativity: Inclusion Of The Undergraduate Student Perspective, Emily Cilli-Turner, V. Rani Satyam, Miloš Savić, Gail Tang, Houssein El Turkey, Gulden Karakok

Mathematics: Faculty Scholarship

Numerous conceptions of creativity exist in the literature; yet these are commonly based on the perspectives of professional mathematicians. Including students’ perspectives in creativity is crucial not only for a more robust picture of what it means to be creative but also to combat damaging dominant narratives about who can be creative. We examined calculus students’ views of mathematical creativity, a group not often considered in the creativity literature, to broaden future considerations of creativity. Interviews with N=55 calculus students across various institutions were conducted. Results show six emergent wide-ranging themes of these students’ creativity views: actions and attitudes, application, …

Math 57: Applied Differential Equations I, John Mayberry

Pacific Open Texts

This book is designed for the fourth semester, “capstone” course in a calculus sequence with an emphasis on modeling with linear differential equations. Students will learn to translate verbal descriptions of physical problems into differential equation models, solve and visualize solutions to differential equations using MATLAB, calculate and investigate the behavior of analytic solutions to linear differential equations, discuss how solutions to differential equations depend on parameters, and interpret solutions to differential equations in the context of applications.

Review Of The History Of Mathematics: A Source-Based Approach (Vol. 2), Part I, Erik R. Tou

Euleriana

Review of The History of Mathematics: A Source-Based Approach (Vol. 2), Part I, by June Barrow-Green, Jeremy Gray, and Robin Wilson. MAA Press, 2022, 330 + xiv pages.

Academic Integration And Self-Regulation Strategies In Precalculus And Calculus I, Kyle Russell Turner

Mathematics Dissertations - Archive

This case study examines the self-regulation strategies and academic integration of first-semester undergraduate students enrolled in precalculus and first-semester calculus at a large urban university in the southwestern United States. I use a sequence of interviews to examine the relationship of mathematics self-efficacy and mathematics identity on students’ use of these strategies. Five interviews occurred during the Fall 2021 semester with three precalculus and six calculus students, and I distributed initial surveys to thirteen first-semester calculus sections and five precalculus sections. To analyze the data, I integrated frameworks from Zimmerman and Pons (1986) and Wolters (1998) on the self-regulation strategies …

Using Teacher Noticing And Video-Mediated Professional Learning To Develop Preservice Teachers’ Knowledge For Teaching The Derivative, Alfred M. Limbere

Theses, Dissertations and Culminating Projects

This study investigated how problem-solving videos can be used in video-mediated professional learning to support secondary preservice mathematics teachers (PMTs) in developing teacher knowledge for noticing student thinking in the context of the derivative concept in calculus. A model of the trajectory of PMTs’ noticing was constructed as six PMTs viewed and analyzed videos of students’ problem solving. At the same time, the nature of video-mediated interactions that were found to be productive in supporting this knowledge development was examined. A design experiment was used as the research methodology. Data was collected from video recordings of eight semi-structured teaching episodes …

Calculus Iii: Under The Influence Of Peer Instruction, Alan Von Herrmann, L. Jeneva Clark

Journal of Humanistic Mathematics

In peer Instruction, students engage with core course concepts and then explain those concepts to one another in small groups. Unlike in lecture format, peer instruction involves every student in the class. In Spring 2019, the first authot began using a modified version of peer instruction in Calculus III classes. He started each class by discussing important Calculus III concepts from three standpoints (the formula, the geometry behind the formula, and the physics behind the formula). During the last 20 minutes of each 50-minute class session, he polled the students using questions in the “Goldilocks Zone” – not too hard …

Accidental World Teacher, Richard Delaware

Journal of Humanistic Mathematics

When the College Algebra and Calculus I video courses I created were posted on my university’s YouTube channel in 2009, I suddenly began to receive dozens of heartfelt emails from students around the world thanking me. Here I tell the story of the creation of those videos and sample the effect they seem to have had over the last decade, as I accidentally became a teacher available to the entire planet.

Asymptotic Dream, Oscar Gonzalez

Journal of Humanistic Mathematics

A love poem about breaching mathematical limits, inspired by the tragic beauty of calculus.

The List: Proverbs For Calculus, Bruce H. Pourciau

Journal of Humanistic Mathematics

Topics chosen from first-year calculus illustrate a number of “sayings” or “proverbs,” the first three, for example, being: be awed, like a child; meaning before truth; and act with intention. Many are proverbs for life as well as mathematics.

Resources For Supporting Mathematics And Data Science Instructors During Covid-19, Eduardo C. Balreira, C. Hawthorne, G. Stadnyk, Z. Teymuroglu, M. Torres, J. R. Wares

Mathematics Faculty Research

In late May of 2020, a few months after the raging COVID-19 pandemic forced university faculty to quickly switch to online teaching, the Associated Colleges of the South (ACS) released a call for grant applications to support working groups "to help faculty within our consortium who will be teaching during the pandemic (e.g., from hybrid courses with some remote/online components to fully remote/online courses; socially distanced face-to-face courses)." We replied to this call and the ACS awarded the six of us (from four ACS schools) a Summer Rapid Response Grant in early June. The grant funded our efforts to create …

Coordinating Stem Core Courses For Student Success, Cristina Villalobos, Hyung Won Kim, Timothy J. Huber, Roger Knobel, Shaghayegh Setayesh, Lekshmi Sasidharan, Anahit Galstyan, Andras Balogh

School of Mathematical & Statistical Sciences Faculty Publications

Research indicates multi-section coordination improves the academic performance of students in STEM education. This paper describes the process of coordination in Precalculus, Calculus 1, and Calculus 2 courses undertaken by a large department that grew from the merger of two institutions through a pilot program, and a project grant. Components introduced in the project courses are documented, including collaborative problem-solving sessions, student learning assistants, Q&A sessions, and additional technology resources. Preliminary data is provided on the impacts of the initiative on student success. The study findings provide a template for coordination, faculty buy-in, and increased student engagement at similar institutions …

Engaging Students Early By Internationalizing The Undergraduate Calculus Course, Chinenye Ofodile

CODEE Journal

Today's world is global. However, despite increasing numbers and diversity of participants in Study Abroad programs, only 10% of U. S. college students get that experience. There is an ever-growing need for students to become aware of and experience other cultures, to understand why others think and act differently. Internationalization is the conscious effort, begun nearly 40 years ago, to integrate an international, intercultural, and global dimension into the purpose, functions, and delivery of post-secondary education.

Albany State University began a Global Program Initiative in the 1990s. In 2016, we extended into mathematics the curriculum innovations of this program. The …

Numerical Integration Through Concavity Analysis, Daniel J. Pietz

Rose-Hulman Undergraduate Mathematics Journal

We introduce a relationship between the concavity of a C2 func- tion and the area bounded by its graph and secant line. We utilize this relationship to develop a method of numerical integration. We then bound the error of the approximation, and compare to known methods, finding an improvement in error bound over methods of comparable computational complexity.

Performance In Calculus Ii For Students In Clear Calculus: A Causal Comparative Study, Ty Mckinney, Rebecca Dibbs

Pursue: Undergraduate Research Journal

Calculus is one of the greatest intellectual achievements of the world and is the main gateway for students that are heading into the fields that will power the economy of the 21st century. However, over 25% of students fail U.S. calculus courses each year and end up changing majors. It is important for educators and researchers to try to improve student success and find ways to increase STEM major retention. The purpose of this study was to compare the performance between students that are in traditional and non-traditional calculus II courses based on their preparation in either traditional or non-traditional …

Supporting Student Success And Persistence In Stem With Active Learning Approaches In Emerging Scholars Classrooms, David Miller, Jessica Deshler, Tim Mceldowney, John Stewart, Edgar Fuller, Matt Pascal, Lynnette Michaluk

Faculty & Staff Scholarship

Over the last several decades, Emerging Scholars Programs (ESPs) have incorporated active learning strategies and challenging problems into collegiate mathematics, resulting in students, underrepresented minority (URM) students in particular, earning at least half of a letter grade higher than other students in Calculus. In 2009, West Virginia University (WVU) adapted ESP models for use in Calculus I in an effort to support the success and retention of URM STEM students by embedding group and inquiry-based learning into a designated section of Calculus I. Seats in the class were reserved for URM and first- generation students. We anticipated that supporting students …

Transitioning To An Active Learning Environment For Calculus At The University Of Florida, Darryl Chamberlain, Amy Grady, Scott Keeran, Kevin Knudson, Ian Manly, Melissa Shabazz, Corey Stone

Publications

In this note, we describe a large-scale transition to an active learning format in first-semester calculus at the University of Florida. Student performance and attitudes are compared across traditional lecture and flipped sections.

Analyzing Applied Calculus Student Understanding Of Definite Integrals In Real-Life Applications, Cody Hood

Graduate Theses, Dissertations, and Problem Reports (ETD)

An individual’s knowledge of definite integrals can range from rote memorization to a strong foundational connection harkening back to its Riemann sum limit definition. In my research, I conducted seven task-based face-to-face interviews with Applied Calculus students. Through the use of real-life examples and guided reinvention, I analyzed ways in which these students, who all initially demonstrated rote memorization, could exhibit a Riemann sum based level of comprehension. This research was conducted in the confines of a student population with definite integral experience, but no formal instruction on limits. My results show that the lack of computational emphasis in class …

Being Active In Research Makes A Person A Better Teacher And Even Helps When Working For A Company, Francisco Zapata, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

At first glance, it looks like being active in research is not necessarily related to a person's success in being a teacher or being a productive company employee -- moreover, it looks like research distracts from other tasks. Somewhat surprisingly, however, in practice, the best teachers and the best employees are actually the ones who are active in research. In this paper, we provide an explanation for this seemingly counter-intuitive phenomenon.