Undergraduate Mathematics Courses Research Articles

Positioned as a burden: Analyzing the participation of multilingual students of color in undergraduate mathematics courses that use groupwork

More college mathematics classrooms are adopting active learning practices like groupwork. These practices typically foreground vocal communication and interpersonal interactions, both of which are mediated by language. Given that undergraduate classrooms in the U.S. are also becoming more racially and linguistically diverse, this study focuses on multilingual students of color, whose learning experiences have often been overlooked. Specifically, this study aims to learn more about the relationship between groupwork and equity in classrooms with linguistic diversity. Using positioning theory, this study analyzes the learning narratives of 26 multilingual students of color about their experiences working with peers in groups. Findings demonstrate that while multilingual students described enacting a range of different positional identities during groupwork, they were more likely to report being positioned by others in deficit ways. Results highlight how the positional identities perceived to be available to students were often constrained by normative classroom Discourses about language, participation, and mathematics.

TEACHER EDUCATION STUDENTS' EXPERIENCES IN UNDERGRADUATE ONLINE MATHEMATICS COURSES

Having been shifted from traditional to online learning due to the community quarantine implementation, it is valuable to determine students' views about taking courses online. This research aimed to explore the teacher education mathematics major students' experiences with taking undergraduate mathematics courses online, precisely the benefits and challenges they encountered, and in what way they dealt with these challenges. The design used in this study is qualitative exploratory research by distributing open-ended questions to selected Bachelor of Secondary Education in mathematics students (N = 76) from a state university in the province of Bulacan, Philippines. Data were explored through thematic coding. The results indicated that among the gains of taking mathematics courses online, the students mostly gained from a convenient study time to access applicable web-based references and resources. In spite of unpredictable internet networks, limited teacher and peer interaction, vague learning content and discussions, some distractions, and a lack of satisfactory learning resources, independent and self-paced study paired and fired with enough self-motivation are suggested. Keywords: mathematics courses, online learning, online mathematics, teacher education, thematic coding

Proposing and testing a model relating students’ graph selection and graph reasoning for dynamic situations

Using a mixed methods approach, we explore a relationship between students’ graph reasoning and graph selection via a fully online assessment. Our population includes 673 students enrolled in college algebra, an introductory undergraduate mathematics course, across four U.S. postsecondary institutions. The assessment is accessible on computers, tablets, and mobile phones. There are six items; for each, students are to view a video animation of a dynamic situation (e.g., a toy car moving along a square track), declare their understanding of the situation, select a Cartesian graph to represent a relationship between given attributes in the situation, and enter text to explain their graph choice. To theorize students’ graph reasoning, we draw on Thompson’s theory of quantitative reasoning, which explains students’ conceptions of attributes as being possible to measure. To code students’ written responses, we appeal to Johnson and colleagues’ graph reasoning framework, which distinguishes students’ quantitative reasoning about one or more attributes capable of varying (Covariation, Variation) from students’ reasoning about observable elements in a situation (Motion, Iconic). Quantitizing those qualitative codes, we examine connections between the latent variables of students’ graph reasoning and graph selection. Using structural equation modeling, we report a significant finding: Students’ graph reasoning explains 40% of the variance in their graph selection (standardized regression weight is 0.64, p < 0.001). Furthermore, our results demonstrate that students’ quantitative forms of graph reasoning (i.e., variational and covariational reasoning) influence the accuracy of their graph selection.

Single-paper meta-analyses of the effects of spaced retrieval practice in nine introductory STEM courses: is the glass half full or half empty?

BackgroundUndergraduate STEM instructors want to help students learn and retain knowledge for their future courses and careers. One promising evidence-based technique that is thought to increase long-term memory is spaced retrieval practice, or repeated testing over time. The beneficial effect of spacing has repeatedly been demonstrated in the laboratory as well as in undergraduate mathematics courses, but its generalizability across diverse STEM courses is unknown. We investigated the effect of spaced retrieval practice in nine introductory STEM courses. Retrieval practice opportunities were embedded in bi-weekly quizzes, either massed on a single quiz or spaced over multiple quizzes. Student performance on practice opportunities and a criterial test at the end of each course were examined as a function of massed or spaced practice. We also conducted a single-paper meta-analysis on criterial test scores to assess the generalizability of the effectiveness of spaced retrieval practice across introductory STEM courses.ResultsSignificant positive effects of spacing on the criterial test were found in only two courses (Calculus I for Engineers and Chemistry for Health Professionals), although small positive effect sizes were observed in two other courses (General Chemistry and Diversity of Life). Meta-analyses revealed a significant spacing effect when all courses were included, but not when calculus was excluded. The generalizability of the spacing effect across STEM courses therefore remains unclear.ConclusionsAlthough we could not clearly determine the generalizability of the benefits of spacing in STEM courses, our findings indicate that spaced retrieval practice could be a low-cost method of improving student performance in at least some STEM courses. More work is needed to determine when, how, and for whom spaced retrieval practice is most beneficial. The effect of spacing in classroom settings may depend on some design features such as the nature of retrieval practice activities (multiple-choice versus short answer) and/or feedback settings, as well as student actions (e.g., whether they look at feedback or study outside of practice opportunities). The evidence is promising, and further pragmatic research is encouraged.

Language demands in undergraduate mathematics courses

This study examines language demands in undergraduate university mathematics classes with a view to better preparing and supporting international and other culturally and linguistically diverse (CLD) students in the United States universities. We observed and analyzed the language demands of 13 entry-level mathematics classes across two large universities. Findings showed that mathematics instruction required extensive and varied listening skills requiring students to listen to long and uninterrupted stretches of sometimes fast paced speech. Technical vocabulary contributed to the language demands, but instructors used both academic and colloquial varieties of language in instruction, interweaving mathematical nouns with phrasal verbs to explain mathematical processes. We suggest a greater focus on development of academic listening skills in English language classes as well as potential collaboration with mathematics campus resources and faculty for additional student support for preparing CLD students for potential future discourse practices.

Promoting elements of mathematical knowledge for teaching related to the notion of assumptions

ABSTRACT Although the notion of assumptions is important in mathematical activity as early as the elementary school, there is limited research on how to help elementary teachers develop mathematical knowledge for teaching related to assumptions. In this paper, we discuss the theoretical foundation and implementation of an intervention that aimed to promote three key elements of this knowledge among prospective elementary teachers. We developed the intervention in a 4-year design experiment that we conducted in an undergraduate mathematics course for prospective elementary teachers. The intervention’s design utilized the notion of productive ambiguity in the context of a deliberately ambiguous task where the role of assumptions surfaced and was reflected upon in purposefully organized ways. We focus on the implementation of the intervention in the last of 5 research cycles of our design experiment to exemplify our theoretical framework and to discuss the promise of the intervention to promote the three targeted elements of knowledge. The approach to promoting mathematical knowledge for teaching that we discuss in the paper offers a paradigmatic case of how teacher educators can use productive ambiguity to design learning opportunities for prospective teachers to intertwine mathematical learning with pedagogical awareness thus developing pedagogically functional mathematical knowledge.

Teaching and Learning of Vertex Coloring Using an Inquiry-Based Approach

Traditional lecturing is the dominant method of teaching mathematics in undergraduate mathematical courses in many countries, whereas active student-centered approaches such as inquiry-based learning have been shown in some instances to be more effective. Most teaching resources (inquiry-based tasks) available at the tertiary level are related to calculus, linear algebra, and differential equations courses. In this paper, we focus on using an inquiry-based approach to explore the teaching and learning of vertex coloring, a common topic in graph theory and undergraduate discrete mathematical courses. We designed seven inquiry-based tasks related to vertex coloring to teach vertex coloring in an undergraduate graph theory course to students majoring in mathematics. In sharing these tasks, describing students’ engagement with them, and discussing how these tasks were perceived by the students and lecturer, we hope to promote using inquiry-based tasks in undergraduate discrete mathematical courses.

Differential equations: students’ reasoning about a problem posing task

This article reports on a qualitative investigation into students' thinking about a differential equations problem posing task; i.e. an initial value problem. Analysis of written and verbal responses to the task indicate that only four of the 34 students who participated in the study were successful in posing problems. Furthermore, only one of the problems they posed is solvable. In general, findings of this study indicate that posing problems involving initial value problems is particularly challenging for students. In light of the multitude of benefits of problem posing in mathematics education, I recommend the inclusion of problem posing tasks in the teaching and learning of undergraduate mathematics courses, including ordinary differential equations.

Encountering ideas about teaching and learning mathematics in undergraduate mathematics courses

We study the ideas about teaching and learning mathematics that undergraduate students generate when they encounter tasks designed to embed approximations of teaching practice in mathematics courses taken by a general population of students. These tasks attend to the dual goals of developing an understanding of mathematics content and an understanding of how teachers provide classroom experiences that foster mathematics learning. The study employs a qualitative, multiple-case study methodology, with four cases bounded by the content areas of abstract algebra, single variable calculus, discrete mathematics, and introductory statistics. The data for the study come from undergraduate students’ written work on mathematical tasks, interviews with a subset of students from each course, and interviews with each instructor throughout the term during which they implemented the tasks. Our findings indicate that students identified the broad applicability of teaching skills (discussed by 32 of the 61 interviewed students), recognized the value of examining hypothetical learners’ mathematical work (discussed by 59 of the 61 interviewed students), and reported empathy for hypothetical learners (discussed by 38 of the 61 interviewed students). These findings persisted across the course content and course levels we studied, leading us to conclude that our findings can transfer to additional mathematics courses in secondary mathematics teacher preparation.

An exploration of course and cohort communication spaces in Discord, Teams, and Moodle

This research examined the impact of supplementing a learning management system, Moodle, with communication tools, Discord and Teams, to support communication in blended and distance undergraduate courses in computer science, information technology, mathematics and statistics at a New Zealand university with well-established use of Moodle. Nineteen students participated in semi-structured interviews. Findings show that adding Discord or Teams increases information and knowledge exchange and helps students to connect with peers and teachers in their courses. Teams was beneficial particularly in settings with formal group work. Discord, which enables both students and staff to set up additional communication channels that are not restricted to course enrolments, was instrumental in connecting students across year levels with peers and alumni. This enabled discussions on course selection, career options and disciplinary topics beyond the course curriculum. Importantly, these beyond-course communication spaces nurtured belonging to wider discipline and study communities. The research establishes the importance of the increased levels of communication by highlighting the effects on student learning and connections to others. Looking beyond the specific tools, the level of formality and the degree of student co-ownership are identified as key factors in supporting the within- and beyond-course communication spaces. Implications for practice or policy: Learning management systems provide valuable course support but do not meet all learning and teaching communication needs, partly due to their formal and university-controlled nature. Educators and students benefit from using strong chat tools with improved information and knowledge exchange. Both educators and students experience enhanced belonging when using collaboration and chat tools. Students value Discord’s informality and student-led characteristics. Educators who use Teams to support formal group work must carefully manage integration with Moodle.

Boundary Transitions Within, Across, and Beyond a Set of Digital Resources: Brokering in College Algebra

We address a problem of promoting instructional transformation in early undergraduate mathematics courses, via an intervention incorporating novel digital resources (“techtivities”), in conjunction with a faculty learning community (FLC). The techtivities can serve as boundary objects, in order to bridge different communities to which instructors belong. Appealing to Etienne Wenger’s Communities of Practice theory, we theorise a role of the instructor as a broker, facilitating “boundary transitions” within, across, and beyond a set of digital resources. By “boundary transition”, we mean a transition that is also a brokering move; instructors connect different communities as they draw links between items in their instruction. To ground our argument, we provide empirical evidence from an instructor, Rachel, whose boundary transitions served three functions: (1) to position the techtivities as something that count in the classroom community and connect to topics valued by the broader mathematics community; (2) to communicate to students that their reasoning matters more than whether they provide a correct answer, a practice promoted in the FLC; (3) to connect students’ responses to mathematical ideas discussed in the FLC, in which graphs represent a relationship between variables. Instructors’ boundary transitions can serve to legitimise novel digital resources within an existing course and thereby challenge the status quo in courses where skills and procedures may take precedence over reasoning and sense-making.

Non-traditional assessments to match creative instruction in undergraduate mathematics courses

Assessment is an issue that is central to the lives of educators and students alike. At all levels, from programmatic to course-specific, instructors strive to assess in a way that accurately measures achievement or progress relative to one’s goals. This paper reports on students’ experiences with alternative assessments in our undergraduate mathematics course for non-STEM (Science, Technology, Engineering and Mathematics) majors. The course, Creative Thinking in Mathematics, designed to fulfil the university’s general education requirement for mathematics, was recently revised and renamed to highlight the role of creativity in mathematics. In the course, students engage in mathematics using creative approaches and consider how these approaches connect to other disciplines, their career aspirations, and important mathematical discoveries throughout history. This shift in course emphasis necessitated revisions to assessments in ways that mirrored the course’s instructional approach. This article describes the alternative assessments, outlines the research study, and reports on the results relative to students’ experiences with these assessments, both prior to and during the course, highlighting a need for instructors to reconsider traditional assessments.

A quantitative exploration of Queer-spectrum students' experiences in introductory undergraduate mathematics courses.

Research into the experiences of students in mathematics with a marginalized sexual identity has remained largely underexplored, relegating these experiences and individuals to remain invisible. In this study, we leverage Queer Theory to address this gap in research by reporting on the experiences of Queer-spectrum undergraduate students (n = 2,454) enrolled in introductory mathematics courses (Precalculus, Calculus 1, and Calculus 2) across the United States. Drawing on student data (n = 24,327) from the Student Post-Secondary Instructional Practices Survey, we examine reported outcomes of math learning experiences and access to learning environments. Overall, within introductory math courses, sexual identity had a significant relationship with accessing learning environments and mathematical learning experiences. Asexual students were among those that reported higher levels of interactions with peers and instructors, a greater sense of community & participation, positive math affect, and were more likely to access learning environments external to the course yet anticipated receiving a lower course grade when compared to straight students. Bisexual students reported lower levels of interactions with instructors, a diminished sense of community & participation, math engagement, positive math affect, and were less likely to access learning environments compared to straight students. These findings suggest that Queer-spectrum students experience mathematical learning in a different manner than their straight peers and in relation to other Queer spectrum identities. Queer-spectrum students with a sexualized identity perceive that mathematics environments are not normative places to be a part of the community, resulting in negative dispositions, reduced interactions, and lower academic success. The impact of sexual identity, although small, contributes to negative indicators of positive mathematical experiences and suggests that the normative discourses in mathematics that are identity neutral, heteronormative, and aligned with binary thinking are differentially impacting Queer-spectrum students.

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.

Metacognitive Assessments for Undergraduate Mathematics Courses in the Time of COVID-19

In this paper, we assess the pedagogical approaches we employ in our US-based undergraduate mathematics courses during the COVID-19 pandemic. Our goal is to share anecdotal evidence on the use of math journals, oral exams, and learning portfolios in the synchronous online mathematics classroom. We reflect on our experience using artifacts of student materials, feedback, and dialogues.

Motivational states in an undergraduate mathematics course: relations between facets of individual interest, task values, basic needs, and effort

According to situated expectancy-value models, motivational variables play an important role in students’ engagement in academic mathematics courses and achievement. However, for academic mathematical learning, only a few studies provide information regarding the fluctuation of task values on different occasions (situation-specificity, state) and the extent to which task values depend on individual characteristics (personal-specificity, trait). Thus, answering these questions provides information about the extent to which adequate situational characteristics, such as relevance interventions, can influence students’ motivation to engage in the learning process. To contribute to this discussion of the situation- and personal-specificity of task values, I surveyed 181 undergraduate students attending an advanced mathematics course. These students reported on their task values and effort three times in four different lessons during a course in linear algebra. I focused on the following three questions: (1) situation-specificity: to what extent do task values fluctuate between situations (and between persons)? (2) predictions of task values: to what extent do personal characteristics and situational perceptions predict task values? (3) predictions of effort: to what extent do personal characteristics and situational perceptions predict effort? These questions were answered by applying linear mixed models. The analysis results indicate that task values fluctuate only slightly in different situations and are strongly predicted by individual characteristics, such as interest in university mathematics, and task values predict effort. The theoretical and practical implications of the results are discussed.

Illustrating Student Mathematics Identities Through a Network of Identity Frameworks

In this paper, we briefly introduce three theoretical frameworks for mathematical identity and why they matter to practitioners teaching undergraduate mathematics courses. These frameworks are narrative identities, communities of practice, and figured worlds. After briefly describing each theory, we provide examples of how each framework can be used to interpret or inform classroom discourses by applying the framework to a selected transcript from a student-centered mathematics project called the Catwalk problem. We summarize how an awareness of students’ forming mathematical identities can be used positively in the undergraduate classroom to help educators support students to form productive mathematical identities.

People-Powered Research and Experiential Learning: Unravelling Hidden Biodiversity

Globally, thousands of institutions house nearly three billion scientific collections offering unparallelled resources that contribute to both science and society. For herbaria alone - facilities housing dried plant collections - there are over 3,000 herbaria worldwide with an estimated 350 million specimens that have been collected over the past four centuries. Digitisation has greatly enhanced the use of herbarium data in scientific research, impacting diverse research areas, including biodiversity informatics, global climate change, analyses using next-generation sequencing technologies and many others. Despite the entrance of herbaria into a new era with enhanced scientific, educational and societal relevance, museum specimens remain underused. Natural history museums can enhance learning and engagement in science, particularly for school-age and undergraduate students. Here, we outline a novel approach of a natural history museum using touchscreen technology that formed part of an interactive kiosk in a temporary museum exhibit on biological specimens. We provide some preliminary analysis investigating the efficacy of the tool, based on the Zooniverse platform, in an exhibit environment to engage patrons in the collection of biological data. We conclude there is great potential in using crowd‐sourced science, coupled with online technology to unlock data and information from digital images of natural history specimens themselves. Sixty percent of the records generated by community scientists (citizen scientists) were of high enough quality to be utilised by researchers. All age groups produced valid, high quality data that could be used by researchers, including children (10 and under), teens and adults. Significantly, the paper outlines the implementation of experiential learning through an undergraduate mathematics course that focuses on projects with actual data to gain a deep, practical knowledge of the subject, including observations, the collection of data, analysis and problem solving. We here promote an intergenerational model including children, high school students, undergraduate students, early career scientists and senior scientists, combining experiential learning, museum patrons, researchers and data derived from natural history collections. Natural history museums with their dual remit of education and collections-based research can play a significant role in the field of community engagement and people-powered research. There also remains much to investigate on the use of interactive displays to help learners interpret and appreciate authentic research. We conclude with a brief insight into the next phase of our ongoing people-powered research activities developed and designed by high school students using the Zooniverse platform.

Competence and autonomous motivation as motivational predictors of college students’ mathematics achievement: from the perspective of self-determination theory

BackgroundApplied Calculus courses serve hundreds of thousands of undergraduates as quantitative preparation and gatekeepers across diverse fields of study. The current study investigated how motivational factors are associated with students’ learning outcomes in Applied Calculus courses from the perspective of self-determination theory—a sound comprehensive motivation theory that has been supported by considerable research in psychology and education. In order to have a nuanced understanding of students’ motivation and learning in Applied Calculus courses, we used three different types of learning measures to investigate students’ mathematics achievement, including course grades, a standardized knowledge exam, and students’ perceived knowledge transferability.ResultsWe tested the relationships between motivational factors and learning outcomes with a multi-semester sample of 3226 undergraduates from 188 Applied Calculus classrooms. To increase the precision of our analysis, we controlled for three demographic variables that are suggested to be relevant to mathematics achievement: gender, minority group status, and socioeconomic status. With a series of multilevel modeling analyses, the results reveal that: (1) competence satisfaction predicts college students’ mathematics achievement over and above the satisfaction of needs for autonomy and relatedness; and (2) autonomous motivation is a more powerful predictor of college students’ mathematics achievement than controlled motivation and amotivation. These findings are consistent across different types of learning outcomes.ConclusionsSelf-determination theory provides an effective framework for understanding college students’ motivation and learning in Applied Calculus courses. This study extends self-determination theory in the field of mathematics education and contributes to the dialogue on advancing undergraduate science, technology, engineering, and mathematics (STEM) education by providing evidence to understand how motivational factors are associated with students’ learning outcomes in undergraduate mathematics courses.

Characteristics of interactive classrooms that first year students find helpful

BackgroundImplementing research-based teaching practices has been repeatedly cited as an important factor for student success in university mathematics courses. Many research-based practices increase the amount of student–student and/or student–instructor interaction. However, some instructors are hesitant to implement such practices because they anticipate their students reacting negatively to experiencing an interactive classroom. As part of a larger project studying introductory undergraduate mathematics courses in the United States, we investigated students’ perceptions of the helpfulness of various classroom characteristics, particularly those that require interaction.ResultsFrom analyzing quantitative student data, we found that students reported interactive classroom characteristics (e.g., group work) as less prevalent than other classroom characteristics (e.g., lecture). Moreover, the students tended to regard characteristics that they reported experiencing often as helpful for their learning. From analyzing qualitative data from student focus groups, we found that students considered several indicators when identifying if a characteristic was helpful for their learning. In particular, students suggested that they can identify a characteristic as helpful for their learning when it supported them in solving assigned problems and understanding why the procedures work, earning good grades, building on their knowledge or applying it in different contexts, and teaching others.ConclusionsThe key finding from our work is that students are likely to view classroom characteristics that they experience more often as more helpful for their learning and are less likely to view characteristics that they rarely experience as helpful for their learning. Students view the characteristics that they regularly experience as helping them to solve problems and understand why the procedures work, earn good grades, build on their knowledge or apply it in different contexts, and teach others. We discuss important implications for practice, policy, and research as it relates to both student and instructor buy-in for increasing interactions in class.