First-year Calculus Course Research Articles

Innovations and Evolution of Large-Enrollment Calculus in Response to the Pandemic

We discuss the impact of the coronavirus pandemic on mathematics instruction at a large, public university with a focus on the impact of first-year calculus courses. At this university, many first-year courses are taught in large lectures with accompanying recitations. Due to previous educational initiatives, we were positioned to leverage the transformation required by the sudden switch to online and hybrid teaching. We made substantial and valuable changes, especially around assessment. We also discuss recent innovations to provide additional pathways and support to students identified as at-risk, including the construction of new course sequences for students intending to take calculus.

Promoting Linear Algebraic Reasoning among Students: Affordances and Challenges

As students transition from the mathematics they learn in school years, including their first-year calculus courses, to the first course in linear algebra, they experience discontinuities in their perspective of what mathematics is. Their propensity to continue applying the same habits of learning in the face of this change leads to failure and frustration. The failure manifests itself in the quality of understanding basic concepts as well as in the lack of linear algebraic reasoning. Instructional treatments applied in my teaching experiments to foster students’ ability to reason linear algebraically resulted in mixed success – some of the treatments were successful, others less so. The latter are accounted for by the structural complexity of the subject matter and students’ background knowledge. The pedagogical approaches offered in this paper are oriented within a particular theoretical framework for the learning and teaching of mathematics, called DNR. Reflections and broader implications are addressed through the lenses of this framework.

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.

Reforming the Teaching and Learning of Foundational Mathematics Courses: An Investigation into the Status Quo of Teaching, Feedback Delivery, and Assessment in a First-Year Calculus Course

Several universities are witnessing an increase in students’ enrolment in mathematics-intensive programmes over the last decades. This increase has come with the price of high failure rates in foundational mathematics courses, which poses challenges to mathematics teaching and learning in higher education. It is therefore inevitable, for some universities, to transform the teaching and learning of mathematics to more student-centred approaches that engage the students mathematically and enhance their success rates. We approach this transformative effort by investigating students’ perception of teaching, feedback, and assessment as a first step in reforming the teaching of a first-year mathematics course at a Norwegian university. The results of both quantitative and qualitative analyses of the data generated using a questionnaire from 107 (80 men) engineering students show that the status quo of teachings offers little support for learning. The teaching is dominated by teacher-led instruction, note-taking, and large pieces of proof which make learning difficult for students during class activities. The results also show that the current structure of the course offers limited formative feedback to students and that the assessment tasks require restructuring to capture students’ time and effort. We discuss the implications of these findings and make some recommendations for improvement.

About the Teaching and Learning of Differentiability for Piecewise Functions in Science Degrees’ First-Year Calculus Courses

In this work, we present an issue we have observed over the last several years in the first year of Spanish science degrees’ (such as mathematics, physics or engineering) calculus courses. It is related to the study of differentiability of piecewise one real variable real functions at a point. We have analyzed how students study the differentiability of piecewise functions at a point, explaining the students’ work and reasoning and commenting about the common misunderstandings we have found. Then, we have researched about how to help the students in their learning process related to this calculus concept, and we have also used several activities or ways to work: student meeting groups to talk about their worries and misunderstandings, mathematical definitions, explanations and theory and also working out specific examples.

Personal Epistemology and Mathematics Performance of First-Year Natural Science Students at the University of Guyana

This research investigated the relationship between epistemological beliefs about mathematics and mathematics performance of first-year students at this University. A convenience sample of 149 students who completed a first-year Calculus course responded to an adaptation of Wheeler’s 2007 epistemological beliefs survey for mathematics (EBSM). This instrument measured four belief dimensions: source of mathematics knowledge, structure and stability of mathematics knowledge, speed and control of mathematics learning, and usefulness of mathematics.

Exploring Educational Change in First-Year Calculus: A Comparative Case Study

This article draws on research conducted in a mandatory, first-year calculus course at the Norwegian University of Science and Technology. By the turn of the century concerns were expressed over high failure rates in calculus, and action was called for to remedy the situation. In the years to come, various measures were introduced, though rather scattered and unsystematic. From 2013 onwards development initiatives in the course became more systematic, ensued by expectations of improved learning outcomes. This study draws on data collected by identical surveys conducted four years apart with a view to similarities and differences in attitudes and behaviours. Contrary to expectations, observed changes turned out to be marginal, if not totally missing, which in turn motivated the search for explanations to design more productive strategies.

Can Students' Attitudes and Behaviors be Changed by Educational Interventions?

This study examined engineering students’ attitudes and behaviors in a first-year Calculus course. Not surprisingly, High School mathematics and physics grades correlated closely with self-reported Calculus grades, and a student survey conducted four years apart demonstrated almost identical attitudes and behaviors despite the introduction of a range of measures aimed to enhance learning. The better the grades, the fairer students deemed it to be, and the less of in-depth learning, the poorer the grades. The higher the ambitions, and the more active and hardworking, the better the grades. Academic success factors included an ability to keep pace with progression, and a commitment to advance learning. The minimal impact of interventions appears as surprising; however, this study brings perspectives to make sense of such data, also capable of producing greater future successes.

DEVELOPMENT OF AN INTERVENTION SCHEME TO ADDRESS LOW RETENTION RATE OF A FIRST-YEAR CALCULUS COURSE – A SYSTEMATIC ANALYSIS OF CURRENT TRENDS

Students in engineering and the sciences often complete their studies in mathematics before they have an opportunity to develop an appreciation for the application of mathematical concepts in their major field. It can be argued that without a solid foundation in mathematics at the calculus level, an engineering or science student will find difficulty in understanding and applying the knowledge involved in upper-level classes. 
 In this study, we examined an Ontario university where the dropout rates could reach as high as fifty percent from a mandatory first-year calculus course and as a response, we would like to develop an intervention mechanism. Using a conceptual framework, we systematically analysed various intervention mechanisms employed by institutions around the world. The framework looks at each intervention strategy and tries to understand how the mechanism identifies who needs intervention, how it is funded, and the steps necessary for a student who would want to receive such an intervention voluntarily. This study will help us to identify key features that are effective in current intervention methods, address the gaps that were observed, and to develop an intervention scheme that addresses high dropout rates from the first-year calculus course.

Small-Scale Learning in a Large-Scale Class: A Blended Model for Team Teaching in Mathematics

We discuss an innovative course model used in first-year calculus courses at a large North American research university, which we refer to as the “Blended Model.” We argue that our model takes the best features of lectures, flipped classrooms, and traditional recitations, and gathers them into a cohesive, robust structure that benefits students’ learning, graduate students’ teaching, and Mathematics departments’ management of resources.

Assessing Peer Leader Skill Acquisition and Group Dynamics in a First-Year Calculus Course

Assessing Peer Leader Skill Acquisition and Group Dynamics in a First-Year Calculus Course

Cameos for calculus: visualization in the first-year course

A thespian or cinematographer might define cameo as a brief appearance of known figure, while gemologist or lapidary might define it as a precious or semiprecious stone. This book presents fifty short enhancements or supplements (the Cameos) for the first-year calculus course in which geometric figure briefly appears. Some of the Cameos illustrate mainstream topics such as the derivative, combinatorial formulas used to compute Riemann sums, or the geometry behind many geometric series. Other Cameos present topics accessible to students at the calculus level but not usually encountered in the course, such as the Cauchy-Schwarz inequality, the arithmetic mean-geometric mean inequality, and the Euler-Mascheroni constant. There are fifty Cameos in the book, grouped into five sections: Part I Limits and Differentiation; Part II Integration; Part III Infinite Series; Part IV Additional Topics, and Part V Appendix: Some Precalculus Topics. Many of the Cameos include exercises, so Solutions to all the Exercises follows Part V. The book concludes with References and an Index.Many of the Cameos are adapted from articles published in journals of the MAA, such as the American Mathematical Monthly, Mathematics Magazine, and the College Mathematics Journal. Some come from other mathematical journals, and some were created for this book. By gathering the Cameos into book we hope that they will be more accessible to teachers of calculus, both for use in the classroom and as supplementary explorations for students.

A Study of Calculus Instructors’ Perceptions of Approximation as a Unifying Thread of the First-Year Calculus

This paper discusses findings from a research study designed to investigate calculus instructors' perceptions of approximation as a central concept and possible unifying thread of the first-year calculus. The study also examines the role approximation plays in participants' self-reported instructional practices. A survey was administered to 279 first-year calculus instructors at higher education institutions throughout the United States. Quantitative and qualitative methods were used to analyze the data gathered. Findings from this research will contribute to what is known about the perceptions and teaching practices of calculus instructors regarding the role of approximation in first-year calculus courses. Research-based findings related to the role of approximation ideas in the first-year calculus could have implications for first-year calculus curricula.

A Short Preparatory Calculus Course: Is it Effective?

In the UTSC Math & Statistics Learning Centre, we designed a 2-week summer preparatory course to help incoming students review background materials that are essential for the success in their first-year calculus courses. This paper discusses the design, implementation, and effectiveness of the course. We examined the effectiveness of the preparatory course through assessing students’ mathematics skills and their confidence of their mathematics skills before and after the course. We also examined the students’ perception and satisfaction with the course. The results show that students demonstrate a higher level of mastery of the fundamental mathematics skills after completing the course; they also tend to be more confident about their mathematics skills. Additionally, students rated the course as very helpful, providing them with a more positive experience than their past experience with mathematics. The course also helped the students getting used to the university learning environment.

Student Success in First-Year University Physics and Mathematics Courses: Does the high-school attended make a difference?

This paper considers school factors that contribute to a successful transition from high school to first-year university Physics courses at the University of British Columbia by employing a two-level hierarchical model. It is assumed that there is a relationship between student performance and the high school they graduated from. It is shown that school location and type affect student performance: students from public schools in the Metro Vancouver area perform better in first year compared to students from independent schools and schools in distant communities. The study also considers rankings of schools based on student performance in first-year university Physics and Calculus courses. These university-based rankings differ significantly (essentially in reverse order) from the well-known Fraser Institute rankings based on measures internal to high schools.

The fundamental theorem of calculus within a geometric context based on Barrow's work

The two main processes of calculus are integration and differentiation. These two processes are intimately related by being reciprocal to each other. This fact constitutes the fundamental theorem of calculus (FTC). In the present article, we explore this important theorem within a geometric context, based on Isaac Barrow's work from the seventeenth century. We argue that presenting and discussing the origins and development of the FTC is beneficial for first-year calculus courses.

Motivation for Achievement and Attitudes toward Mathematics Instruction in a Required Calculus Course at the Norwegian University of Science and Technology

This study from the Norwegian University of Science and Technology (NTNU) examines students’ learning goals and attitudes toward mathematics in a first-year calculus course in undergraduate engineering education. Achievement motivation research using the Achievement Goal Questionnaire (AGQ) is advanced from current literature with two additions: (1) a course specific context using introductory college calculus students, and (2) participation of Norwegian students. Pre- and posttest measures of attitudes indicate that students do change learning goals over time, unfortunately opposite to the instructors’ aspirations. A significant increase in “Mastery Avoidance” and “Work Avoidance” was accompanied with a drop in “Mastery Approach” and “Performance Approach”. Variables such as value, motivation and enjoyment decreased along with a significant drop in self-confidence.

The Basic Principle of Calculus?

A simple partial version of the Fundamental Theorem of Calculus can be presented on the first day of the first-year calculus course, and then relied upon repeatedly in assigned problems throughout the course. With that experience behind them, students can use the partial version to understand the full-fledged Fundamental Theorem, with further assigned problems leading them through that process.

Some counterexamples for your calculus course

Abstract We discuss some examples and counterexamples that are used in first-year calculus courses to illustrate the interconnections between various function classes. Particular emphasis is put on the two-parameter family of functions (1.1) which provides an unexhaustable source of counterexamples.

University students’ difficulties in solving application problems in calculus: Student perspectives

This paper reports on the results of an observational parallel study conducted simultaneously at 2 universities — one each in New Zealand and Germany. It deals with university engineering students’ difficulties in the formulation step of solving a typical application problem from a first-year calculus course. Two groups of students (54 in New Zealand and 50 in Germany) completed a questionnaire about their difficulties in solving the problem which was set as part of a mid-semester test. The research endeavoured to find reasons most of the students could not use their knowledge to construct a simple function in a familiar context. It was neither lack of mathematics knowledge nor an issue with the context. The students’ difficulties are analysed and presented along with their suggestions on how to improve their skills in solving application problems.