Collegiate Mathematics Research Articles

Reflecting on Professional Development Opportunities: Links Between Conceptions of Mathematics Graduate Teaching Assistants and Their Self-Efficacy

This study examines the perceptions of first-year, mathematics graduate teaching assistants (GTAs) participating in a six-week summer course designed to deepen their knowledge of collegiate mathematics teaching practices before being assigned to instruct undergraduate students in the subsequent fall semester. Through hybrid deductive-inductive thematic analysis, GTAs’ written reflections were analyzed and matched against changes in their self-efficacy assessed along two conceptual dimensions: self-improvement and stimulation of student learning. Results suggest that GTAs’ conceptualization of teaching practice informs changes in self-efficacy along both conceptual dimensions, with efficacy in self-improvement changing more than that of their ability to stimulate student learning. Therefore, the strength of the efficacy changes may be moderated by features of the professional development course. We explore defining features of the professional development course employed in this study, describing implications for the education and development of novice collegiate mathematics instructors, and the potential to optimize change along both conceptual dimensions of self-efficacy.

Listening for Common Ground in High School and Early Collegiate Mathematics

Listening for Common Ground in High School and Early Collegiate Mathematics

Collegiate Mathematics Observation Protocols: Current Characteristics and Desired Foci

Collegiate Mathematics Observation Protocols: Current Characteristics and Desired Foci

Collegiate mathematics teaching in proof-based courses: What we now know and what we have yet to learn

There has been significant growth in the literature base exploring questions of teaching in undergraduate mathematics. In this paper we synthesize the literature on the teaching of proof-based undergraduate mathematics, drawing on 104 published reports from a range of countries and research traditions. We primarily differentiate the papers into those which explore lecture-based pedagogy and student-centered pedagogy. For each type of instruction, we focus on three categories of findings from the literature: description of instruction, instructor beliefs and rationales, and the relationship between instruction and students (cognitive, participatory, affective, and equity oriented). Much is known about the enactment of lecture-based teaching, including instructors’ cognitive and affective goals. The student-centered literature focuses on tensions and challenges implementing curricula with a greater focus on participatory goals. Overall, there are few studies that attempt to link instructors’ classroom activity and students’ learning. Similarly, attention to equity is relatively lacking in the extant research.

Using a constraint-based expert model to provide step-level feedback for user-inputted mathematics equations

Studies across a variety of educational fields have shown the efficacy of feedback on student performance and learning. Web-based homework is a common feature of secondary and collegiate mathematics courses to provide such feedback. While web-based homework provides often instantaneous feedback to students as they complete assignments, the feedback is limited to the correctness of the final answer or to a limited number of pre-programmed common mistakes. Intelligent Tutoring Systems have the ability to provide step-by-step feedback and guidance tailored to individual students, however authoring content for such systems is generally considered a huge barrier to their wider use. This paper presents an alternative system, called SANYMS (Show and Name Your Math Steps), which combines the wide-scale usability of web-based homework with a Constraint-Based Intelligent Tutoring System Expert Model based on the open-source computer algebra system Maxima. With the integration of Maxima, SANYMS can provide step-by-step feedback to students constructing their own solutions to a wide variety of algebraic equations of their choice without the need for extensive content authoring inherent in many Intelligent Tutoring Systems.

One is the Loneliest Number: Groupwork within Linguistically Diverse Classrooms

ABSTRACT Collegiate mathematics instructors are moving away from traditional lecture-based teaching practices to accommodate the pedagogies that support more active student learning. As part of the movement, groupwork has been increasingly implemented. At the same time, mathematics classrooms have become increasingly diverse and also in the distribution of students' primary languages. In this report, we examine two students' experiences with groupwork in a collegiate mathematics course. What commonly binds these two students is that they each found themselves to be the only member of their respective groups who uses a different primary language from the other group members. After presenting the experiences, we provide instructional implications for group formation in linguistically heterogeneous classrooms.

Supporting Student Success and Persistence in STEM With Active Learning Approaches in Emerging Scholars Classrooms

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 in courses in the calculus sequence, including Calculus I, would support URM Calculus I students in building learning communities and serve as a mechanism to provide a strong foundation for long-term retention. In this study we analyze the success of students that have progressed through our ESP Calculus courses and compare them to their non-ESP counterparts. Results show that ESP URM students succeed in the Calculus sequence at substantially higher rates than URM students in non-ESP sections of Calculus courses in the sequence (81% of URM students pass ESP Calculus I while only 50% of URM students pass non-ESP Calculus I). In addition, ESP URM and ESP non-URM (first-generation but not URM) students succeed at similar levels in the ESP Calculus sequence of courses (81% of URM students and 82% of non-URM students pass ESP Calculus I). Finally, ESP URM students’ one-year retention rates are similar to those of ESP non-URM students and significantly higher than those of URM students in non-ESP sections of Calculus (92% of ESP URM Calculus I students were retained after one year, while only 83% of URM non-ESP Calculus I students were retained). These results suggest that ESP is ideally suited for retaining and graduating URM STEM majors, helping them overcome obstacles and barriers in STEM, and increasing diversity, equity, and inclusion in Calculus.

Exploring Mathematics of the Sociopolitical Through Culturally Relevant Pedagogy in a College Algebra Course at a Historically Black College/University

In collegiate mathematics, college algebra continues to be a barrier to graduation for students (specifically non-science, mathematics, engineering, and science majors). Each year, nearly half of enrolled students struggle to “pass” this course with a grade of C or better (Herriott, 2006). Using innovative constructed lessons geared towards African American students, this research study was designed to investigate the effects of a sequence of such lessons grounded in the principles of culturally relevant pedagogy on students enrolled in an introductory college algebra course at a historically Black college/university. Using critical race theory as a lens, along with culturally relevant pedagogy, this study explored students’ abilities to apply mathematics to address contentious and present-day sociopolitical problems through eight in-depth semi-structure student interviews. Further, findings also suggest the need for collegiate mathematics instruction to have more emphasis on cultural components to build students’ sociopolitical consciousnesses, because this is integral in helping students be able to think critically and use mathematics in their everyday lives. Students in this experimental course were able to discuss difficult issues, such as the pervasiveness of racism in America (DeCuir & Dixson, 2004) and the importance of cultural identity for African American students (Martin, 2009).

Summative Portfolios in Undergraduate Mathematics Courses

A summative portfolio is a project completed at the end of the semester that gives a student an opportunity to look at a course as a whole. This study investigates whether collegiate mathematics students gain a deeper understanding of course material through the portfolio project, via an analysis of the students’ work, and a survey of students’ opinions.

How elementary and collegiate instructors envision tasks as supportive of mathematical argumentation: A comparison of instructors’ task constructions

We examine how instructors at different grade levels proposed mathematical tasks to support students’ engagement in constructing viable arguments and critiquing others’ reasoning. During small group interviews with pairs of practicing elementary and collegiate mathematics instructors, they created tasks intended to support students’ ability to create and critique mathematical arguments. From the anslysis of their written and verbal work, we found that although all instructors requested explanations in their created tasks, there were key differences in the nature of reasoning-and-proving expected. Grade 1 instructors tended to focus on empirical justifications and procedural explanation; Grade 3 and 4 instructors included similar requirements, but had more emphasis on soliciting rationales; and college instructors required students to ultimately develop more formalized arguments, consistently including requests for conjectures and generalizations. The findings provide evidence for key differences in argumentation norms at specific levels in schooling, and implications for research, teaching, and curricular design.

From course design to presentations of proofs: How mathematics professors attend to student independent proof reading

Undergraduate mathematics students are expected to spend a substantial amount of time outside class studying proofs independently. Yet, little is known about how such expectations shape instruction in proof-oriented courses. This paper investigates this issue in a case study of an experimental teaching strategy, designed by a mathematics professor, for scaffolding student independent proof reading. On the basis of interviews and classroom observations we scrutinize the instructor’s course image – his envisioning of the teaching and learning activities, inside and outside class, throughout the course. We examine a representative case of proof presentation through the lens of the instructor’s course image. Our analysis reveals complex relationships between various teaching and learning activities in the course, and demonstrates that unpacking these relationships can provide valuable insights into specific teaching practices. Findings suggest that the concept of course image can be useful in enhancing our understanding of mechanisms underlying collegiate mathematics instruction.

Mathematics and Mourning: Textbook Burial and Student Culture Before and After the Civil War, 1853–1880

In nineteenth-century America, students buried their mathematics books. This practice consistently celebrated the milestone of passing through collegiate mathematics, yet it changed due to national events. This article considers the case of Bowdoin College, where students buried their books differently before and after the Civil War. Antebellum, they observed a complex “Burial of Calculus” with songs, parades, and mock prayers. Postbellum, students personified their books as a woman, placing stones marked “Anna” on the textbooks’ graves. Using archival investigations of students' pamphlets and textbooks, this paper argues that these changes resulted from the war's effects on education as well as changing attitudes toward death. Both the antebellum and postbellum rituals communicated understandings of mathematics and academic achievement, as connected through a mock funeral ritual. Through investigating these connections, this paper asserts the importance of student practices for our understanding of Civil War era education.

Implementing a flipped instructional model in college algebra: profiles of student activity

ABSTRACTFlipped instruction is increasing in popularity, however research that moves beyond descriptions of its implementation in mathematics classes is lacking. We sought to better understand how students taking an introductory college algebra course used the resources provided within a flipped instructional model and how students viewed such resources as useful to their learning. Data from course activity records and student usage surveys were used to characterize the frequency of student access to course resources (e.g. video lectures and comprehension assessments) and students' perceptions of the usefulness of these resources. Students naturally fell into one of four categories with respect to video usage and success in the course. Students who employed videos as intended were successful in the class. Students rated watching videos and working on homework in class as the activities that best supported their learning. Students did not perceive group work and in-class presentations as helpful. Our findings have implications for other instructors implementing flipped instruction. We argue that while flipping collegiate mathematics courses has great potential, instructors need to explicitly attend to the design of the in-class components in order to maximize student learning.

Professional Development in Teaching for Mathematics Graduate Students

The purpose of this article is twofold: to give a sense of the current lay of the land in the preparation of mathematics graduate student teaching assistants (TAs) and to describe the collegiate mathematics education research base informing the next generation of college mathematics instructor preparation. We anchor discussion in three common types of TA preparation programs, each represented in one of the quotes above. Notably, the first quote represents a sink or swim experience that is becoming rare in US PhD-granting mathematics departments. Preparation of TAs for their instructional roles has blossomed in the last twenty years. The second quote is representative of current practices in many departments. The third quote illustrates the activities in innovative departments that are already implementing best practices suggested by research in collegiate mathematics education: sustained professional growth about teaching and learning. To highlight the challenges and benefits of spending time paying attention to teaching, we provide information from postsecondary and related secondary-level educational research of several types. This includes basic and applied educational research that identifies good instructional practices, examines experiences TAs bring with them to teaching, provides frameworks for the structuring of TA preparation, and gives insight into the kinds of mathematics-specific and teaching-specific knowledge that needs to be developed among TAs. And, once a program for supporting TAs to learn and grow as instructors is put in place, evaluation research explores the implementation of efforts to improve TAs’ teaching and the related impacts on undergraduate student learning. We close with promising practices and sketch anticipations for the future of the field of related research. Though not common thirty years ago, today most doctorateand master’s-granting institutions provide some kind of TA preparation for teaching [4a]. The content for this professional development often comes from mathematicians offering their collective wisdom from practical Jessica M. Deshler is assistant professor of mathematics at West Virginia University. Her email address is deshler@ math.wvu.edu.

Making Materials Based on TeX and CAS/DGS — Reports on CADGME 2012 Conference Working Group —

TeX has become one of the most popular tools for editing teaching materials or textbooks in collegiate mathematics education, since it enables usual mathematics teachers to easily produce high-quality mathematical documents. Its capabilities for visualization and computation are fairly limited, so that many teachers simultaneously use various software such as Computer Algebra System (CAS) or Dynamic Geometry Software (DGS) to generate and display high-quality mathematical artworks. In this paper, we discuss the conditions under which the graphics generated by CAS/DGS mostly enhance the educational effect of teaching materials edited by TeX. As a result, the use of a CAS macro package for generating high-quality graphics in TeX documents is proposed.

Alternate Trajectories: Women Moving into Mathematics Education

While only about one-third of each year’s doctoral graduates in mathematics are women, about two-thirds of the doctoral graduates in mathematics education are women. This article reports on the results of a qualitative investigation into the nature of the graduate school-related experiences of women in collegiate mathematics education doctoral programs (Toney, 2008). Eight women with advanced mathematics degrees were interviewed. Each woman chose to move into a collegiate mathematics education doctoral program in a mathematics department. The two-interview protocol explored and extended a framework about women doctoral mathematics students’ experiences suggested by the research of Herzig (2002, 2004a, 2004b), Hollenshead, Younce, and Wenzel (1994), and Stage and Maple (1996). Results support an addition of three new categories to this framework: self as scholar, "my teaching," and future possible self. Concluding remarks include suggestions for future research and new directions for practice, graduate program development, and faculty recruitment.

The History of the Undergraduate Program in Mathematics in the United States

This article describes the history of the mathematics major and, more generally, collegiate mathematics, in the United States. Interestingly, the Mathematical Association of America was organized about 100 years ago, around the same time that academic majors came into existence.

HBCUs Propel African American Male Mathematics Majors

While researchers have examined the educational experiences of African American male students at historically black colleges and universities (HBCUs), researchers have neglected to adequately hone in on the mathematics experiences of African American male students at these institutions. Studies reporting on their mathematics experiences usually call attention to African American male students’ lack of mathematics preparation for collegiate mathematics and/or the difficulties these students experience in collegiate mathematics courses. This qualitative study, however, highlights the importance of HBCUs in producing successful African American male mathematics majors. Findings provide evidence that HBCUs provide supportive structures, mechanisms, and people, especially African American male mathematics professors, who contribute to the mathematical success of African American male mathematics majors.

A Lesson Learned from Outer Billiard Derivatives

As students move onward and upward through collegiate mathematics they are often impressed by the power of advanced techniques (aren’t they?), techniques allowing problems that were previously difficult or near-impossible to be solved with relative ease. For example, once students have learned the residue theorem in complex analysis they are able to elegantly evaluate integrals that were unwieldy in calculus. Among practicing mathematicians there is also a tendency to look to new or powerful results when trying to unlock a problem within one’s own research. This article presents a new theorem concerning derivatives within the outer billiard dynamical system, and in so doing serves as a reminder that higher-powered mathematics are not always needed and do not necessarily produce more satisfying proofs. Sometimes it is beneficial to keep elementary approaches in mind.

Collegiate mathematics teaching: An unexamined practice

Though written accounts of collegiate mathematics teaching exist (e.g., mathematicians’ reflections and analyses of learning and teaching in innovative courses), research on collegiate teachers’ actual classroom teaching practice is virtually non-existent. We advance this claim based on a thorough review of peer-reviewed journals where scholarship on collegiate mathematics teaching is published. To frame this review, we distinguish between instructional activities and teaching practice and present six categories of published scholarship that consider collegiate teaching but are not descriptive empirical research on teaching practice. Empirical studies can reveal important differences among teachers’ thinking and actions, promote discussions of practice, and support learning about teaching. To support such research, we developed a preliminary framework of cognitively oriented dimensions of teaching practice based on our review of empirical research on pre-college and college teaching.