Undergraduate Mathematics Curriculum Research Articles

Discussion on the construction of vocational undergraduate mathematics curriculum system

The country has established a pilot project for undergraduate vocational education, proposing new requirements for mathematics courses. Taking the curriculum system positioning and construction planning as the benchmark, this paper starts from the early stage of preparation, reconstructs the curriculum, and establishes a brand-new mathematics curriculum system. The final framework consists of a core curriculum of general education that strengthens the foundation and a disciplinary foundation platform curriculum that highlights the cultivation of applied abilities. Different courses are designed for different majors to adapt to the new challenges of vocational undergraduate education.

AN ASSESSMENT-DIRECTED, STUDENT-CENTERED, AND MASTERY-BASED MODEL OF TEACHING AND LEARNING IN MATHEMATICS

This is a descriptive article illustrating a multi-year research study in mathematics education of college students in the U.S. It focuses on the first two years in the undergraduate mathematics curriculum. The study presents the theory, practice, and results of an innovative instructional program (Keystone model) applied to algebra classes at the college. The research investigated two areas of student learning outcomes: performance in mathematics and improvements in students’ concentration skills. Under the application of this program, students not only achieved significant outcomes in mathematics, as evidenced by formative and summative assessments but also improved their concentration skills, as demonstrated in the gains in national reading comprehension tests. In the following, we present, in extensive detail, the elements of the Keystone approach and its implementation in a learning environment, such as a college mathematics classroom.

Learning Mathematics from the Master: A Collection of Euler-based Primary Source Projects for Today’s Students, Part II

This article and its prequel together highlight a set of nine classroom-ready projects that draw on the remarkable writing of Leonhard Euler (1707--1783) as a means to help students develop an understanding of standard topics from today’s undergraduate mathematics curriculum. Part of a larger collection of primary source projects intended for use in a wide range of undergraduate mathematics courses, these projects are freely available to students and their instructors. We provide a general description of the pedagogical design underlying these projects, more detailed descriptions of the individual projects themselves, and instructions for obtaining downloadable copies for classroom use.

The Value of Computation: A Few Illustrative Examples

ABSTRACT This paper presents a series of basic computational problems that are mathematically and/or graphically appealing, and provides an idea of places one might go in trying to understand what is happening, integrating mathematics, computation, and graphics. The real point of this paper is to make a case, through those examples, for computation as an early and integral part in the undergraduate mathematics curriculum. By computation we mean the ability to rapidly and easily program exploratory calculations, and as well display the results with a versatile suite of graphics capabilities. The problems used here, and the philosophy behind them, derive from a MATLAB-based course in computation, programming, and mathematical applications that the author has taught for over 10 years.

Learning Mathematics from the Master: A Collection of Euler-based Primary Source Projects for Today’s Students, Part I

This article and its sequel will together highlight a set of nine classroom ready projects that draw on the remarkable writing of Leonhard Euler (1707–1783) as a means to help students develop an understanding of standard topics from today’s undergraduate mathematics curriculum. Part of a larger collection of primary source projects intended for use in a wide range of undergraduate mathematics courses, these projects are freely available to students and their instructors. We provide a general description of the pedagogical design underlying these projects, more detailed descriptions of the individual projects themselves, and instructions for obtaining downloadable copies for classroom use.

Formalizing Galois Theory

ABSTRACT We describe a project to formalize Galois theory using the Lean theorem prover, which is part of a larger effort to formalize all of the standard undergraduate mathematics curriculum in Lean. We discuss some of the challenges we faced and the decisions we made in the course of this project. The main theorems we formalized are the primitive element theorem, the fundamental theorem of Galois theory, and the equivalence of several characterizations of finite degree Galois extensions.

Enriching Undergraduate Mathematics Curriculum with Computer Science Courses

Traditional mathematics curriculum faces several issues nowadays. The gap between course materials and students’ real-life mathematical experiences, the scattering of knowledge in different courses, and the lack of mathematics applications to other subjects all hinder the learning of students. The emerg-ing trends in data science, machine learning, and artificial intelligence also impel higher education to enrich and refine mathematics education. In order to better incubate students for future, the experience of enriching undergrad-uate mathematics curriculum with computer science courses is introduced in this study. The curriculum is designed and implemented for students who major in applied mathematics to better stimulate the learning, participation, exercise, and innovation. It provides students with comprehensive theoretical and practical knowledge for the challenges and industrial requirements now-adays. Evaluations, major findings, and lessons learned from three refined courses are discussed for more insight into the following deployment and re-finement of the curriculum.

Estimating the Sensitivity of Fitted Parameters to Perturbations of Data with Calculus

This article provides conceptual ideas, data, and exercises, for integrating original sources of recent, state of the art, world-class life science research in the undergraduate mathematics curriculum and classroom. To this end, this article shows how one of the main goals of calculus in the life sciences, fitting parameters to data and assessing the reliability of the fitted parameters, fits in existing mathematics courses at levels from basic to multivariable calculus. The material for multivariable calculus explains methods of computation of quantitative criteria of “significance” that are currently the subject of scientific controversy. This article thus provides resources to strengthen students' mastery of calculus before, or in, mathematical modeling, probability, and statistics.

The Lorenz Curve in the Classroom

The Lorenz curve and Gini index have great social relevance due to concerns regarding income inequality. However, their discussion is limited in the undergraduate statistics and mathematics curriculum. This article outlines how to increase the educational potential of Lorenz curves as an application in both the calculus class and introductory probability classroom. We show how calculus and probability techniques can be used to obtain not only the Gini index, but also a variety of other statistical measures from the Lorenz curve, provided the mean is known. The measures discussed include the median, and various measures of dispersion.

Using a Superfactorial Problem to Provide Extended Problem-Solving Experiences

ABSTRACT The purpose of this paper is to describe the use of a problem involving superfactorials (a specific product of factorials) that provides an in-depth and comprehensive mathematical experience, encompassing skills that mathematicians view as tantamount to exploration. The problem is easily accessible and fosters creativity and perseverance, thereby allowing participants to practice standard steps in developing a mathematical argument: from the problem, to simplifications, then conjecture, and finally proof. We present a brief solution to this problem and then discuss how the experience in trying to solve this problem may evolve during a professional development session for teachers. By providing teachers with extended time to investigate a single problem, we aim to provide professional development opportunities beyond the standard mathematical content or instructional practices. This problem also makes a welcome addition to any undergraduate mathematics curriculum and could be used in a mathematics for liberal arts course, capstone course, or any course with a focus on developing problem-solving skills and the process of doing mathematics.

Demystifying Proofs Through Structured Interaction: A Case Study of One Instructor’s Teaching in an Undergraduate Analysis Course

The concept of proof is central to every undergraduate mathematics curriculum, but it is also a difficult concept for university students to understand and for university instructors to teach. Prior studies have contributed useful research knowledge about the nature and sources of undergraduate students’ difficulties with proof, but they have paid less attention to the development of effective instructional practices to enhance student learning of proof. As a step towards this direction, this paper makes a contribution to the limited body of case studies of promising or potentially effective teaching practices in undergraduate mathematics by reporting a case study of an instructor’s teaching, locally characterized as “effective,” in an undergraduate analysis course at a leading Indian university. The instructor did not deviate from the so called “Definition-Theorem-Proof” (DTP) format that is followed in most proof-oriented university mathematics courses, but her teaching presented a set of features that, collectively, form a rather innovative teaching practice at the undergraduate level. Specifically, our case study shows that, even in the context of a rather crowded university classroom, proof can be demystified for students through structured interaction between the instructor and the students, that is, an interactive, conversational style of proof instruction invoking the participation of students. This is based on a solid foundation in symbolic logic at the very outset and on a significant time investment in definitions being explained in depth using informal language, visual aids, and real-life analogies. In addition to contributing to existing images of potentially effective teaching practices in undergraduate mathematics, this paper draws attention to the Indian educational context that has had little representation in international forums of mathematics education research thus far.

Editorial

This special issue of MSOR Connections is based on contributions to a workshop, ‘Programming in the undergraduate mathematics curriculum’, which took place at Middlesex University on 27th June 2019, and other submissions received on the same theme.

Programming in Groups: developing industry-facing software development skills in the undergraduate mathematics curriculum

Programming is increasingly becoming an expected graduate skill for mathematics students. We argue in this article that programming should be given the same priority as any other graduate skill. Given the practical and philosophical constraints placed on undergraduate mathematics curricula, however, we acknowledge the difficulty in introducing, in a meaningful way, many of the core ideas of programming. We therefore present a case study of a second year course on an undergraduate mathematics programme that introduces Object Oriented Programming and aspects of software design, as well as key practical skill such as version control. We will argue that group assessment in this context is a more natural setting for students to be working and reflects more closely the experience of programming in industry; furthermore, it serves as a convenient platform to introduce students to aspects of software design and practical programming considerations. We will present an example of the type of assessment that can be used and how Version Control Systems like Git can be used to give students a more realistic experience of programming with the advantage of allowing tutors and other group members to track student work.

Harder things will stretch you further: helping first-year undergraduate students meaningfully engage with recent research papers in probability and statistics

Abstract Undergraduate research increasingly features in university mathematics degrees. Despite this, research papers are used infrequently in mathematics teaching, and this is especially the case for first-year undergraduates. Mathematical subjects are more likely than other STEM disciplines to pinpoint cognitive difficulty as the principal reason for not exposing undergraduate students to research papers. In this paper, we test whether first-year students can engage effectively with research papers. We describe an intervention that exposes first-year, first term undergraduate students to current research in probability and statistics by asking them to read a research paper and summarize it for a general readership following an interview with the paper’s author. Our findings show that the activity introduced students to new fields of knowledge and helped to develop a clearer understanding of scientific process, leading to a heightened sense of personal satisfaction at engaging closely with current research. We argue that structured reading of research papers can lead to productive and rewarding engagement with difficult content, recent and current research and with research processes and that this should make us reconsider the role of research papers in the undergraduate mathematics curriculum.

Integrating Programming Across the Undergraduate Mathematics Curriculum

We have integrated computer programming instruction into the required courses of our mathematics major. Our majors take a sequence of four courses in their first 2 years, each of which is paired with a weekly 75-minute computer lab period that has a dual purpose of both computationally exploring the mathematical concepts from the lecture portion of the course, while simultaneously teaching programming fundamentals. Building on this foundation, we give significant programming assignments to our upper-division students, requiring them to regularly use and apply their programming skills to investigate the mathematical topics.

A Vision for Projects Across the Mathematics Curriculum

We consider the use of projects in math courses as a mechanism for promoting coding, communication, and interdisciplinary application of math skills. Final projects play an important role, but we also discuss several alternate types of projects. We describe a model that incorporates projects at all stages across the undergraduate mathematics curriculum. Finally, we provide samples of projects that we have used in selected courses.

Mini-research Experience in Simple Linear Regression

Within the undergraduate mathematics curriculum, the topic of simple least-squares linear regression is often first encountered in multi-variable calculus where the line of best fit is obtained by using partial derivatives to find the slope and y-intercept of the line that minimizes the residual sum of squares. A markedly different approach from linear algebra, which could also be introduced in multi-variable calculus, obtains the regression line by vector projection. The latter viewpoint offers elegant proof of the equation relating the total, explained and unexplained variations. Consideration of data with the same regression line and correlation opens the door for a “mini-research experience” (MRE). A sequel MRE gives rise to an open Research Experience for Undergraduates topic to analyze reflection sequences and a fundamental connection between complex analysis and regression analysis. A few general guidelines and basic goals for MREs are included for those whose main interest is in undergraduate research.

Klein’s Plan B in the Early Teaching of Analysis: Two Theoretical Cases of Exploring Mathematical Links

We present a theoretical approach to the problem of the transition from Calculus to Analysis within the undergraduate mathematics curriculum. First, we formulate this problem using the anthropological theory of the didactic, in particular the notion of praxeology, along with a possible solution related to Klein’s “Plan B”: here, re-linking the theory of Analysis with practical knowledge from Calculus. We explore two cases based on this approach: (1) the contribution of Vector Analysis to the foundations of trigonometric functions, and (2) establishing the ties between the proof of a basic theorem in Fourier Analysis and the computation of elementary infinite series. These two cases, including small-scale experiences, illustrate the necessity, importance and possibilities of new didactical approaches aiming to help student to integrate mathematical theories and practices which are otherwise taught separately.

Computer programming in the UK undergraduate mathematics curriculum

abstractThis paper reports a study which investigated the extent to which undergraduate mathematics students in the United Kingdom are currently taught to programme a computer as a core part of their mathematics degree programme. We undertook an online survey, with significant follow-up correspondence, to gather data on current curricula and received replies from 46 (63%) of the departments who teach a BSc mathematics degree. We found that 78% of BSc degree courses in mathematics included computer programming in a compulsory module but 11% of mathematics degree programmes do not teach programming to all their undergraduate mathematics students. In 2016, programming is most commonly taught to undergraduate mathematics students through imperative languages, notably MATLAB, using numerical analysis as the underlying (or parallel) mathematical subject matter. Statistics is a very popular choice in optional courses, using the package R. Computer algebra systems appear to be significantly less popular for compulsory first-year courses than a decade ago, and there was no mention of logic programming, functional programming or automatic theorem proving software. The modal form of assessment of computing modules is entirely by coursework (i.e. no examination).

Understanding and advancing graduate teaching assistants’ mathematical knowledge for teaching

Graduate student teaching assistants (GTAs) usually teach introductory level courses at the undergraduate level. Since GTAs constitute the majority of future mathematics faculty, their image of effective teaching and preparedness to lead instructional improvements will impact future directions in undergraduate mathematics curriculum and instruction. In this paper, we argue for the need to support GTAs in improving their mathematical meanings of foundational ideas and their ability to support productive student thinking. By investigating GTAs’ meanings for average rate of change, a key content area in precalculus and calculus, we found evidence that even mathematically sophisticated GTAs possess impoverished meanings of this key idea. We argue for the need, and highlight one approach, for supporting GTAs to improve their understanding of foundational mathematical ideas and how these ideas are learned.