Undergraduate Mathematics Classroom Research Articles

The Harp Project: Collective Learning at the Intersection of the Mathematical and Musical Arts

With this paper, we share an activity for the undergraduate mathematics classroom called The Harp Project that leverages the aesthetic nature of both the mathematical and the musical arts. This project was conceived as a STEAM/PBL project with the added feature that it was carried out in pieces by an entire class. Anecdotal evidence suggests that the uncommonly high level of students' interest and engagement in the experience and the communal feel of their participation can be attributed to this collective quality of the project's design. For this reason, we invite others to consider this curricular experience as it appears viable for supporting efforts to cultivate a broader population of students with an affinity for mathematics.

Flipped Pre-Calculus for Engineers: An Active Learning Course Transformation

ABSTRACT As part of a departmental reform project, a large University converted a “Precalculus College Mathematics” lecture-based course for 97 students into an active learning flipped classroom for first-year engineering students. The curriculum was designed specifically to provide applications of pre-calculus in various engineering subjects. This paper outlines the course evolution, design features, ongoing adjustments made to the course, and quantitative data from the common examination and course evaluations to support the instructional design. The paper concludes with a reflection on which aspects of the learning environment helped facilitate successful learning and guidance for future implementations of a large-scale flipped undergraduate mathematics classroom.

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.

Introduction to the Special Issue on Implementing Mastery Grading in the Undergraduate Mathematics Classroom

This editorial serves as an introduction to the Special Issue on Implementing Mastery Grading in the Undergraduate Mathematics Classroom. We provide a broad introduction to mastery grading and describe articles in this special issue that provide a wide range of implementations of mastery grading in a variety of undergraduate mathematics classrooms. We conclude by identifying additional resources available to new practitioners, and describing ideas for future work related to mastery grading. We hope this special issue will serve as a resource for both experienced practitioners and those who wish to begin using mastery grading.

Inquiry and Gender Inequity in the Undergraduate Mathematics Classroom

Our field has generally reached a consensus that active-learning approaches improve student success; however, there is a need to explore the ways that particular instructional approaches affect various student groups. We examined the relationship between gender and student learning outcomes in one context: inquiry-oriented abstract algebra. Using hierarchical linear modeling, we analyzed content assessment data from 522 students. We detected a gender performance difference (with men outperforming women) in the inquiry-oriented classes that was not present in other classes. We take the differential result between men and women to be evidence of gender inequity in our context. In response to these findings, we present avenues for future research on the gendered experiences of students in such classes.

The Effects of Instructional Approach and Social Support on College Algebra Students’ Motivation and Achievement: Classroom Climate Matters

College algebra has been noted as a critical course in post-secondary institutions because it serves as a gateway for major selection and college completion. Combinatorial topics like repeatable permutations are often overlooked in K-12 and undergraduate curricula. Likewise, students’ achievement and motivation are affected by the type of classroom climate created in the undergraduate mathematics classroom. Inquiry-based mathematical education (IBME) is a viable instructional approach because of its focus on community meaning-making of the mathematical content. However, lecture-style approaches still dominate post-secondary mathematics classrooms even though they have been criticized for their focus on procedural knowledge and disinviting environment. Therefore, the purpose of this quasi-experimental study was to test the effects of instructional approach (i.e., lecture-style vs. IBME) and social support (i.e., absence or presence) on undergraduate student motivation and achievement of combinatorial mathematics. Findings indicated that intentional social support-building – regardless of pedagogical method – had the strongest effects on students’ perceived autonomy-support, competence and achievement. Although no differing pedagogical effects were discovered (most likely due to the one-time implementation of the lesson formats), the findings provide evidence for the necessity of community-building efforts -- an aspect of education that is often overlooked in the undergraduate mathematics classroom.

Using Analytics to Support Instructor Reflection on Student Participation in a Discourse-Focused Undergraduate Mathematics Classroom

This paper describes a method for using instructional analytics to support changes in teaching practice. The study took place in a discourse-focused undergraduate mathematics classroom taught by an experienced instructor. Using the classroom observation tool EQUIP and social network surveys, we generated quantitative data on patterns of student participation in this classroom that were presented to the instructor at the end of the semester. Our analysis focuses on how the instructor made sense of these analytics and ultimately changed his teaching practices in a future semester. This paper provides insight into how three data sources can be triangulated—classroom participation, student experiences, and teacher perspective—to better understand discourse-based participation in undergraduate mathematics. We also discuss the implications of using this methodology for faculty professional development.

Key memorable events: A lens on affect, learning, and teaching in the mathematics classroom

This paper proposes a theoretical construct termed Key Memorable Events (KMEs): classroom events that are perceived by many students as memorable and meaningful in support of their learning, and are typically accompanied by strong emotions, either positive or negative. Though the approach taken in this paper is theoretical, the construct is illustrated and supported by empirical data collected in the undergraduate mathematics classroom. I address the potential value of the construct, and demonstrate its fit with existing constructs. Theoretical and pedagogical implications are discussed in terms of lesson design, data analysis, and our conceptualization of student learning, whilst enriching the existing metaphor of learning as a trajectory.

Chalk it up to experience: using chalkboard paint to create mathematical manipulatives

In this article, we give two examples of creating portable chalkboards using chalkboard paint for students to use during cooperative learning. This provides a creative method for professors to facilitate active learning in the undergraduate mathematics classroom.

Teaching Mathematics with Women in Mind

W e both work at institutions that have been awarded NSF ADVANCE grants. ADVANCE is a program that aims to develop systematic approaches to supporting academic women in science, technology, engineering, and mathematics (STEM) disciplines. ADVANCE looks at universities and asks, “What support can universities provide to help women faculty members thrive in STEM?” We took that question to heart and asked ourselves, “What support can we provide in our undergraduate mathematics classrooms to help women students thrive in mathematics?” We decided to take a hard look in the mirror. This meant examining our own approaches to teaching and confronting the limits of our own education and preparation. Trained as mathematicians, we found our preparation for understanding how gender affects undergraduate mathematics classroom environments lacking. We turned to resources from psychology, sociology, and women’s and gender studies to better understand our own teaching. Our personal goal—it might not be yours—is to understand how gender bias influences our mathematics teaching and to explore concrete steps we can take to reduce the effects of that bias. We’ve asked ourselves why our classrooms should look different from the classrooms in which we were undergraduate students. After all, we are both women who were successful undergraduate mathematics majors and went on to earn Ph.D.’s in mathematics. Call it survivor’s guilt: we both know that we were able to “fit in” in mathematics when many other women could not. We could say, “Other women should do what we did and learn to fit in.” Instead, we are asking, “What should we do differently as mathematics instructors so that more women, who wouldn’t otherwise be successful, will be and will persist in the study of our subject?” We have to make a few stipulations. First, we believe teaching can be taught. Though trained as applied mathematicians, we both now do research in and teach mathematics education. Our work in mathematics education is guided by the belief that great teaching can be taught. This is contrary to what popular culture has us believe: often teachers are characterized as “born to teach.” Sometimes great teachers believe it too. There is something to that—in the absence of preparation for teaching, there are those who will become better because they experiment, reflect, and revise their own teaching. They do this on their own and end up keeping the best of their experiments and learning from the ineffective ones. We don’t mean to diminish the greatness of those who have figured out how to become great teachers on their own; we want to understand what they do in classrooms and teach others to do the same. As we explore changes to make in a classroom to be responsive to gender, we are thinking of explicit teaching choices. Second, we believe there is inherent good in giving more people access to undergraduate mathematics. This is a value we stipulate. You might not share that value, but if that’s the case, then you and we are aiming at different things. We’ve thought a lot about what kinds of good come from access to mathematics, and we believe that what might resonate most with our students is the access to fulfilling, higher-paying jobs that comes with advanced technological knowledge. Some may argue that this is not why we teach mathematics, but it Jessica M. Deshler is assistant professor of mathematics at West Virginia University. Her email address is deshler@ math.wvu.edu.

Knowledge Needed by a Teacher to Provide Analytic Scaffolding During Undergraduate Mathematics Classroom Discussions

Using case study analysis and a cognitive theoretical orientation, we examine elements of knowledge for teaching needed by a mathematician to orchestrate whole-class discussions in an undergraduate mathematics classroom. The instructor, an experienced teacher and mathematics researcher, used an inquiry-oriented curriculum to teach a differential equations course for the first time after teaching it with traditional lecture methods for many years. Examples of classroom teaching and interview data demonstrate that, despite having extensive teaching experience and possessing strong content knowledge, some instructors may still face challenges when trying to provide analytic scaffolding to move whole-class discussions toward a lesson's mathematical goals. We also hypothesize several component practices necessary for the successful use of analytic scaffolding. Our analysis focuses on the relationship between the instructor's pedagogical content knowledge and specialized content knowledge and his capacity to enact these component practices during whole-class discussions.

Proofs and refutations in the undergraduate mathematics classroom

In his 1976 book, Proofs and Refutations, Lakatos presents a collection of case studies to illustrate methods of mathematical discovery in the history of mathematics. In this paper, we reframe these methods in ways that we have found make them more amenable for use as a framework for research on learning and teaching mathematics. We present an episode from an undergraduate abstract algebra classroom to illustrate the guided reinvention of mathematics through processes that strongly parallel those described by Lakatos. Our analysis suggests that the constructs described by Lakatos can provide a useful framework for making sense of the mathematical activity in classrooms where students are actively engaged in the development of mathematical ideas and provide design heuristics for instructional approaches that support the learning of mathematics through the process of guided reinvention.

Storying Joanne,1 an undergraduate mathematician

In this article two different accounts are juxtaposed. In one, we use a variety of texts to narrate the story of Joanne, a woman undergraduate student of mathematics. Like many of our mature students Joanne came to the university with a ‘non‐traditional’ academic background. We describe how Joanne developed as a learner of mathematics and connect this to our ways of working in the undergraduate mathematics classroom. We believe that our pedagogy is unusual outside (some) school classrooms and suggest it allows our students to develop positive ‘disciplinary relationships’. In the other, we grapple with the issues raised by telling other people’s stories especially when we are also characters within it. Our intention has been that, in interweaving these two threads, each helps us understand more about the other.

Using an Undergraduate Geometry Course to Challenge Pre-Service Teachers' Notions of Discourse.

This investigation uses classroom discoursein an undergraduate mathematics course tochallenge pre-service secondary mathematicsteachers' notions about mathematical discourse,what it might resemble in the classroom, andhow its various forms can be cultivated byclassroom teachers. The research setting was arequired geometry course taught by the author.Eleven pre-service mathematics teachers intheir junior or senior year of an undergraduate program participated. Results indicated participants made a transition toward an image of discourse as an active process by which students use the collective knowledge of their peers to build mathematical understanding and developed in their ability to participate in such discourse.This awareness, along with participants'analyses of their own habits of discourse asclassroom teachers, prompted shifts in theirprojected image of the role of discourse intheir future practices of teaching. Resultssuggested further that the undergraduatemathematics classroom (as opposed to themethods classroom) offers a powerful and uniqueforum in which pre-service secondary teacherscan practice, articulate, and collectivelyreflect on reform-minded ways of teaching.

Technology for deciding the convergence of series

The use of technology to motivate the study of calculus is very popular in today's undergraduate mathematics classroom. This article presents some new ways of using a spreadsheet in teaching foundations of the theory of infinite series. The approach is based on a spreadsheet's ability to model inequalities of the Bolzano‐Cauchy type and provides students with a kind of visual imagery of the process of convergence (divergence). The main concern of the work is the issue of a computer visualization and exploration that lead to a conjecture. For treating slowly converging series the method of dynamic visualization is proposed which illustrates the usefulness of a spreadsheet in rather delicate explorations. It is shown that a computer laboratory equipped with spreadsheets can be a good teaching environment to help students develop the basic concepts of infinite