We welcome presentations that share findings from research in undergraduate mathematics education, including qualitative or quantitative empirical studies and theoretical discussions. We are open to a wide range of topics, including but not limited to: the teaching and learning of particular concepts, teaching at the undergraduate level, student cognition, and effective classroom interventions.
Organizer:
Elise Lockwood
Contact:
Ekaterina (Katya) Yurasovskaya*, Seattle University
Seattle University has a long history and a solid institutional structure for implementing academic service-learning in its courses. For the present study, we developed a Precalculus course with a service-learning component, allowing university students to work in the tutoring labs at a local middle school, an immigrant assistance center, and a community college, and to tutor algebra prerequisites to middle-school students and to adults returning to complete their GED diploma. One of the primary goals of the project was to improve basic algebra skills of the student tutors by explaining foundational material to others 2-3 hours per week over the course of the quarter. Through weekly pedagogical diaries, the student tutors analyzed the source and nature of mathematical successes or misconceptions of their own students. Review of the final exams via a special rubric revealed a significant reduction in the number of fundamental mistakes between the Precalculus section with the service-learning component and the control section of the same course. In our talk, we discuss the collaboration with community partners, the course structure and the key components that enhanced student
learning, and academic and non-academic benefits to the participants.
Research indicates that in order for students to be able to successfully interpret and flexibly model with definite integrals, they must conceptualize the integral as “adding up the pieces” of a quantity, rather than as (a) a symbolic template, (b) an anti-derivative, or even just (c) an area under a curve. Task-based interviews with students in my recent experimental calculus class indicate that the “adding up the pieces” conceptualization of $\int_a^b f(x) dx$ requires the student to be able to view the “f(x)dx” as a small bit of the total quantity. This understanding in turn relies on being able to treat “dx” as an increment of x, which has a size, across which the quantity to be summed accumulates (at a rate of f(x)). I argue that this incremental understanding of dx was also crucial in allowing Leibniz to develop calculus where his predecessors (e.g. Cavalieri and Torricelli), who used indivisibles that have no “width,” did not.
This talk considers what students attend to as they first encounter R3 coordinate axes and are asked to graph y = 3. Graphs are critical representations in single and multivariable calculus, yet findings from research indicate that students struggle with graphing functions of more than one variable. We found that some students thought y = 3 in R3 would be a line, while others thought it would be a plane. In creating their graphs, students attended to equidistance, parallelism, specific points, and the role of x and z. Students’ use of these ideas was often generalised from thinking about the graphs of y = b equations in R2. A key finding is that the students who thought the graph was a plane always attended to the z variable as free.
The Cauchy Property is an important characterization of convergent sequences in complete metric spaces. Students were observed reflecting on the nature of Cauchy Sequences on R, and were then prompted to generalize the definition of a Cauchy sequence into more abstract settings. Their generalizations were implemented into various metrics to eventually be leveraged for the re-invention of the abstract metric definition. This talk will explore some of the students' initial generalizing activity and will describe some of their cognitive schemes developed through interactions with various notions of distance.
Time: Saturday, April 2, 2016 - 11:15
Room Number: STAG 262
Emily Cilli-Turner*, University of Washington Tacoma
Mathematicians and mathematics educators agree that proof has many different roles in mathematics beyond that of verifying the truth of a statement. For instance, some proofs can not only show that a statement is true, but also explain why it must be true. However, students may not appreciate these other functions of proof as they are not explicitly taught in the classroom. This report outlines an inquiry-based teaching intervention in an introduction to proof course and its impacts on students' appreciation of other functions of proof. Results show that exposure to inquiry pedagogy changed students' conceptions of the functions of proof and increased their recognition of the explanatory and communication power of proof.
During creation of the Group Concept Inventory (GCI), we discovered that the initial question related to isomorphism adapted from Weber & Alcock (2004) (Are Q and Z isomorphic?) contained a number of hidden complexities related to student understanding of isomorphism. We developed two new questions to attempt to better parse apart student conceptions around the topic. One question targeted cardinality and the second structural sameness. The questions were piloted with group theory students across the country. We analyzed the data using a thematic analysis approach. Additionally, we conducted five follow-up interviews to further make sense of student reasoning related to isomorphism. In this presentation, we focus on student use of formal and informal definitions, their ability to explain the relationship between structural sameness and the formal definition, and their flexibility addressing non-familiar properties.
The multiplication principle ("MP") is fundamental to combinatorics, underpinning many standard formulas and providing justification for counting strategies. Given its importance, the way it is presented in textbooks is surprisingly varied. In this talk, we identify key elements of the principle and present a categorization of statement types found in a textbook analysis. We incorporate excerpts from a reinvention study that shed light on how students reason through key elements of the principle. We conclude with a number of potential mathematical and pedagogical implications of the categorization.
While researchers have found that students at a variety of levels struggle to solve counting problems correctly, listing has been shown to be a potentially effective remedy to student difficulties. Motivated by its importance in helping students solve counting problems, this presentation describes a research study conducted to investigate attitudes that undergraduate students have about listing. This study found that undergraduate students seem to agree that listing is a useful activity, but they may not have a strong understanding of how to productively use a list of outcomes to solve counting problems. Implications for pedagogy and future research directions are discussed.
Ninety-eight elementary and middle school prospective teachers participated in a study focusing on integer addition and subtraction while enrolled in an introductory mathematics content course emphasizing number concepts and operations. Across two academic semesters, the prospective teachers posed 784 stories for integer addition and subtraction number sentences. Of these, 108 of the stories included the context of temperature. The stories about temperature were analyzed for mathematical correctness, consistency, realism, and problem types. A framework for different problem types for integers was modified. Other complexities for posing these types of stories, such as realism of the temperature stories, will be discussed.