Undergraduate Calculus Research Articles

Extending a technique for integrating quotients of linear combinations of sines and cosines

ABSTRACT We discuss a non-routine technique to solve some classes of integrals, which could enrich the syllabus of a first-year undergraduate calculus course. We begin with the class from which the inspiration originates: integrals of quotients of linear combinations of sines and cosines. Subsequently, we discuss a generalization, dealing with quotients formed by a linear combination of two arbitrary functions on the denominator and a linear combination of their derivatives on the numerator. Finally, for a variation, we consider integrals of quotients of linear combinations of three functions: exponentials, sines, and cosines.

The pandemic as a catalyst for rethinking active learning practices in technology intensive instructional environments

Even though K-12 mathematics educators have embraced the role of graphing calculators and computers, student use of technology in tertiary mathematics has been slow to implement in the USA context. Enter March 2020, a global health crisis, and the shift to online and remote learning. The immediate and dramatic cessation of in-person modes of instruction was experienced at all levels of education and in all disciplines. With respect to tertiary mathematics, student use of instructional technology shifted from optional use of homework websites, digital discussion boards and flipped classrooms to an absolute necessity to support student participation in mathematics courses. Our research group has been studying the process of departmental change towards active learning in the undergraduate calculus sequence. During the COVID-19 pandemic, we worked with twenty-three university mathematics departments in the United States as part of a Networked Improvement Community (NIC). During the pandemic, we observed the adaptation of instructional practices to sustain student interaction in remote virtual environments. This paper highlights changes observed within our NIC regarding attempts to sustain active learning during the global pandemic. In addition, a NIC can provide the necessary community for grappling with challenging problems at any time.

Teaching the Chain Rule to Manage Cognitive Load: A Colorful Approach

Based on our work teaching undergraduate Calculus courses, we offer insight into teaching the chain rule to reduce cognitive load for students. A particularly difficult topic for students to grasp, problems likely arise due to student struggles with the concept of function and, particularly, function composition relative to when they first encounter the chain rule. We suggest that, by incorporating the intentional use of color in instruction, we can highlight the particular aspects of the symbols we want students to focus on, thereby reducing cognitive load and early difficulties solidifying the topic.

Digital Instructional and Pedagogical Gaps in Mathematics Education during COVID-19: APOS Theory Based Laboratory Methodology in Higher Education

The COVID-19 pandemic globally transformed the face to face teaching and learning into digital online teaching and learning. The uptake and integration of digital technology in mathematics education have been gradual for many decades. However, the COVID-19 pandemic has reshaped the digital teaching and learning and made it an imperative and safe remote instruction and pedagogy of the day in mathematics education and education in general. We explored the uptake and use of virtual mathematics laboratory methodology for undergraduate Calculus teaching and learning as a remote instruction and pedagogy gap revealed during the COVID-19 infections prevention and containment period in Zimbabwe. The main objective of the study was to identify digital instructional and pedagogical gaps that existed in addressing students' mathematical misconceptions in Calculus learning in using the APOS theory and a virtual mathematics laboratory. Data was collected from 160 undergraduate students taking Calculus courses via APOS theory mediated digital mathematics laboratory for a semester. Results revealed that the undergraduate students who took the Calculus courses via the digital mathematics laboratory methodology performed better than those who used to take the same courses using didactical approaches. This COVID-19 crisis ushered in digital teaching and learning experiences that clearly spelt out the existence of digital instructional and pedagogical gaps in mathematics education.

Engaging Students Early by Internationalizing the Undergraduate Calculus Course

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 result has engaged students in a serious way, both in mathematical modeling and in cultural research. We have introduced students to new skills of research and presentation. For the past few years we have offered one section of Calculus II in traditional mode and one in internationalized mode, and we have compared results. In this article, we give details of the process and highlight the success of the program. We end with more recent examples from Spring 2020.

Spatial Training and Calculus Ability: Investigating Impacts on Student Performance and Cognitive Style

Undergraduate calculus is a foundational mathematics sequence that previews the sophistication students will need to succeed in higher-level courses. However, students often struggle with concepts in calculus because they are more abstract and visual than those in other foundational mathematics courses. Additionally, women continue to be underrepresented in the STEM fields. This study builds on previous work indicating a malleability in spatial ability by testing whether improvement occurs in students’ spatial and mathematics ability after implementing spatial training in calculus courses. The researchers also measured associations between spatial training and self-reported cognitive style. While spatial training did not significantly improve calculus and spatial skills, the researchers measured impacts on the psychological correlate of cognitive learning style. This is important as non-spatial cognitive preferences can result in students not recognizing the dynamic information encapsulated in graphical images, which is a large component of calculus. These results may have practical implications for teaching calculus at the undergraduate level and may, with further research, help to narrow the gender gap in the STEM fields.

The Collaborative Development of Active Learning at Loyola University Chicago as Part of the SEMINAL Network

ABSTRACT In the context of national collaborative efforts to increase the prevalence of active learning strategies in undergraduate precalculus and calculus classrooms, we discuss the context, strategy, and processes used at one university to create change within a mathematics department. We highlight the use of a unified active learning approach and the importance of collaborative practices as leverage for departmental change. We conclude with lessons learned from the first year of implementation.

A precise, simple and general Basic Le Châtelier Principle based on elementary calculus: What Le Châtelier had in mind?

The Le Châtelier Principle is one of the most important concepts in chemistry, and it has been the topic of many publications over the years. However, its meaning and application are often fraught with misunderstanding and confusion. As a suggested replacement, we present herein a precise general statement that we call the Basic Le Châtelier Principle (BLCP), which is in keeping with a common thread in Le Châtelier’s original statements. The BLCP is formulated as a consequence of well-known properties of a simple but general optimization problem, which elevates its range of application beyond chemistry to any phenomenon governed by such an optimization principle. We show applications of the BLCP to simple example problems in economics and in physics, in addition to the usual chemistry problems,. Following a brief outline of Le Châtelier’s original statements, we formulate the BLCP, which incorporates the notion of “de signe contraire” (of opposite sign), common to all his statements. It arises by abstracting the chemical reaction equilibrium problem (CREP) in the single-reaction case to the general problem of minimizing a differentiable function $$f(x;\{p_j\})$$ , where x is the single independent variable and $$\{p_j\}$$ is a set of parameters. The BLCP arises from an exact expression for the dependence of the sign of the incremental change in the optimal solution $$x^*$$ on the sign of the incremental change in a parameter, which is derived using techniques taught in an early undergraduate calculus course. When translated back to the CREP, this yields unambiguous expressions for the sign of the incremental change in the equilibrium reaction extent, $$\xi ^*$$ , arising from an incremental change in each of T, P, the initial species amounts, $$\{n_i^0\}$$ , and the standard reaction free energy change, $$\varDelta G^{\Box }$$ . Special emphasis is placed on the requirement that f must satisfy a positive definite second derivative condition, for which we present a proof in the case of multiple reactions in an ideal solution model system. We also briefly consider the extension of the single-variable BLCP derived herein to the case of multiple independent variables and to finite parameter perturbations.

Active learning strategies with positive effects on students’ achievements in undergraduate mathematics education

The paper describes active learning strategies used in undergraduate college algebra and business calculus courses. There are a variety of active learning strategies described in the literature. We wanted to implement a few that together, satisfy the key characteristics of active learning strategies. The active learning strategies described in this paper are, interactive presentation style, group-work with discussion and feedback, volunteer presentations of solutions by groups, raise students’ learning interest towards specific topics, involve students in mathematical explorations, experiments, and projects, and last but not least, continuous motivation and engagement of students. We analyse the relationship between the average results of 27 college algebra and business calculus sections and the effects of the level of use of these active learning strategies. We demonstrate that the application of these strategies has a positive effect on the average results of the sections and the passing rates of the students.

Invariance of the Length and the Area of Cycloids

To make a marked point on the boundary of a circular wheel of radius 1 trace the curve known as a cycloid, let this wheel roll along an arbitrary fixed curve called the base curve. There are two cycloids, each of which is drawn on one side of the base curve and share cusps with each other. Consider the two arcs joining the same pair of adjacent cusps, one arc from each cycloid. T. M. Apostol and M. A. Mnatsakanian showed that the sum of the lengths of these arcs is independent of the shape of the base curve. They also showed that the sum of the areas bounded by two cycloids is independent of the shape of the base curve. Their remarkable proof does not use integral calculus. We generalize these results to the general trochoid case. Our proof is based on simple undergraduate calculus.

The Case for Raabe’s Test

SummaryAmong the techniques for determining the convergence of a series, Raabe’s test remains relatively unfamiliar to most mathematicians. We present several results relating to Raabe’s test that do not seem to be widely known, making the case that Raabe’s test should be featured more prominently in undergraduate calculus and analysis courses. In particular, we demonstrate that Raabe’s test may be viewed as an implicit comparison with a p-series, in the same manner that the ratio test and the root test constitute an implicit comparison with a geometric series. Moreover, Raabe’s test can sometimes simplify the process for determining conditional convergence.

Calculus students’ quantitative reasoning in the context of solving related rates of change problems

ABSTRACTDespite the increasing amount of research on students’ quantitative reasoning at the secondary level, research on students’ quantitative reasoning at the undergraduate level is scarce. The present study used task-based interviews to examine 16 high-performing undergraduate calculus students’ quantitative reasoning in the context of solving three related rates of change problems-two geometric and one non-geometric. A qualitative analysis of verbal responses and work written by the students when solving the three problems revealed that 15 students created and used diagrams to support their quantitative reasoning in the geometric problems, and that the creation of these diagrams helped the students to solve the two problems. In addition, seven students exhibited lack of facility with the product or quotient rule of differentiation. We argue that the use of diagrams in calculus instruction at the undergraduate level could play an important role in supporting students’ quantitative reasoning in topics that require students to make sense of relationships between quantities such as related rates of change problems. Furthermore, we argue that helping students develop greater facility with rules of differentiation such as the product or quotient rule could increase students’ success in many calculus application problems, including related rates of change problems. Directions for future research are included.

Motivating Calculus Through a Single Question

The Corvette Problem is a nonroutine investigation, one that provides an authentic context to explore many interrelated calculus ideas. In our years of sharing this problem with students and colleagues, we have found additional topics from the calculus curriculum to have rich interpretations in this environment. The Corvette Problem contextualizes much of the material in first-semester undergraduate calculus by way of unifying theorems and definitions into a powerful mosaic of ideas.

WAYS OF USE OF COMPUTER ALGEBRA SYSTEMS IN THE TEACHING OF ADVANCED SECTIONS OF MATHEMATICS

In this article, possible ways of use of computers for the teaching of advanced sections of mathematics that traditionally belong to undergraduate curricula, namely elements of calculus, number theory and abstract algebra are considered. Use of computer technologies can help also to implement such approaches as genetic method and the use of various modes of representation in education. According to cultural-historical theory of L. S. Vygotsky, computer technologies can be considered as the tool for the construction of concepts in the process of learning. The most appropriate for teaching advanced mathematics are such computer algebra systems as Maple, Mathematica, and various systems of dynamic geometry. We will consider the possibilities of Geogebra for the work with functions at the initial stages of undergraduate calculus courses, namely for the work with concepts of limits and derivatives of functions.

Investigating classroom implementation of research-based interventions for reducing stereotype threat in calculus

ABSTRACTMathematics is a discipline that exists within a social context, a context that influences how the subject is taught and learned. Women's performance in mathematics has been shown to be vulnerable to stereotype threat, and psychologists have developed interventions aimed at reducing the effect of stereotype threat on women's mathematics performance. We applied interventions, shown to be effective at reducing gender stereotype threat in mathematics in a lab setting, in undergraduate calculus classrooms. We found that these interventions had no detectable effect in the classroom setting, leading to two important cautions about the application of lab-based studies in authentic settings. First, lab settings differ from classroom settings in fundamental ways, limiting the ability to generalize inferences from lab studies to classroom settings, a caution that is more well-known amongst researchers seeking to replicate studies than it is amongst practitioners seeking to apply studies in practice. Second, when seeking to apply lab-based studies in classroom settings, researchers must be aware that relying on research constructs (such as sample size and significance) masks the practical measure of a meaningful sample size, which is roughly the number of students in a particular instructor's course at a given time.

Forged in failure: engagement patterns for successful students repeating calculus

Although there is extensive research on attrition in gatekeeper courses and students’ cognition about calculus concepts, there is one population in introductory calculus that remains understudied: those who failed their initial course and chose to repeat it rather than change majors. These students can provide insight into overcoming poor mathematics affect and major persistence. This case study follows eight students repeating calculus from their second try at undergraduate calculus until they graduated or left the university; six graduated with either a mathematics major or mathematics minor. While participants identified several reasons for their success in the repeated course (processing their initial failure, having a better instructor in the repeated course, and participating regularly in the formative assessments), only participation in formative assessment led to the long-term cognitive and behavioral engagement required for long-term success.

A calculus course in knowledge feedback format

This paper presents a design of an undergraduate calculus course which attempts to achieve higher student activity and teacher-student exchange, allowing the teacher to adapt more to the present student group and to increase students work time. This is done by switching between examinations in small format and examination-free, open mathematical dialogue occasions – two learning-enhancing opposites that reinforce each other during the course. In the examination-free dialogue, students solved problems during lectures every week, where the quality of the student-teacher dialogue is essential both for dissection of properties of mathematics and for students’ willingness to continue participating. Thus, students regularly practiced verbal presentation and reflection in mathematics.

Simple Fractal Calculus from Fractal Arithmetic

Non-Newtonian calculus that starts with elementary non-Diophantine arithmetic operations of a Burgin type is applicable to all fractals whose cardinality is continuum. The resulting definitions of derivatives and integrals are simpler from what one finds in the more traditional literature of the subject, and they often work in the cases where the standard methods fail. As an illustration, we perform a Fourier transform of a real-valued function with Sierpi\'nski-set domain. The resulting formalism is as simple as the usual undergraduate calculus.

From Smileys to Smileycoins: Using a Cryptocurrency in Education

This paper describes a cryptocurrency to reward students for their studies. The currency bears the apt name Smileycoin or SMLY and is used within the tutor-web online learning platform. In order to make the SMLY attractive to students several approaches have been used, including support from companies whose services can be purchased for SMLY. The paper describes the use of the SMLY as a reward mechanism in a large undergraduate calculus course, including student adoption, student use of SMLY, coinbase use for education in low-income areas, and response to abuse.

Assessing Preservice Teachers’ Mathematics Cognitive Failures as Related to Mathematics Anxiety and Performance in Undergraduate Calculus

Assessing Preservice Teachers’ Mathematics Cognitive Failures as Related to Mathematics Anxiety and Performance in Undergraduate Calculus