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4:20 – 4:50 pm

Session 5 – Preliminary Reports

Marquis A

Use of strategic knowledge in a mathematical bridge course: Differences between an undergraduate and graduate

Darryl Chamberlain Jr. and Draga Vidakovic

The ability to construct proofs has become one of, if not the, paramount cognitive goal of every mathematical science major. However, students continue to struggle with proof construction and, particularly, with proof by contradiction construction. This paper is situated in a larger research project on the development of an individualÕs understanding of proof by contradiction in a transition-to-proof course. The purpose of this paper is to compare proof construction between two students, one graduate and one undergraduate, in the same transition-to-proof course. The analysis utilizes Keith WeberÕs framework for Strategic Knowledge and shows that while both students readily used symbolic manipulation to prove statements, the graduate student utilized internal and flexible procedures to begin proofs as opposed to the external and rigid procedures utilized by the undergraduate.



Marquis B

Classifying combinations: Do students distinguish between different types of combination problems?

Elise Lockwood, Nicholas Wasserman and William McGuffey

In this paper we report on a survey designed to test whether or not students differentiated between two different types of problems involving combinations – problems in which combinations are used to count unordered sets of distinct objects (a natural, common way to use combinations), and problems in which combinations are used to count ordered sequences of two (or more) indistinguishable objects (a less obvious application of combinations). We hypothesized that novice students may recognize combinations as appropriate for the first type but not for the second type, and our results support this hypothesis. We briefly discuss the mathematics, share the results, and offer implications and directions for future research.



Marquis C

A case study of developing self-efficacy in writing proof frameworks

Ahmed Benkhalti, Annie Selden and John Selden

This case study documents the progression of one non-traditional individualÕs proof-writing through a semester. We analyzed the videotapes of this individualÕs one-on-one sessions working through our course notes for an inquiry-based transition-to-proof course. Our theoretical perspective informed our work with this individual and included the view that proof construction is a sequence of (mental, as well as physical) actions. It also included the use of proof frameworks as a means of initiating a written proof. This individualÕs early reluctance to use proof frameworks, after an initial introduction to them, was documented, as well as her later acceptance of, and proficiency with, them. By the end of the first semester, she had developed considerable facility with both the formal-rhetorical and problem-centered parts of proofs and a sense of self-efficacy.



Grand Ballroom 5

Results from the Group Concept Inventory: Exploring the role of binary operation in introductory group theory task performance

Kathleen Melhuish and Jodi Fasteen

Binary operations are an essential, but often overlooked topic in advanced mathematics. We present results related to student understanding of operation from the Group Concept Inventory, a conceptually focused, group theory multiple-choice test. We pair results from over 400 student responses with 30 follow-up interviews to illustrate the role binary operation understanding played in tasks related to a multitude of group theory concepts. We conclude by hypothesizing potential directions for the creation of a holistic binary operation understanding framework.



City Center A

Online calculus homework: The student experience

Andrew Krause

The MAA advertises that the online homework system WeBWorK is used successfully at over 700 colleges and universities, and the institution selected for my study has implemented WeBWorK universally across all calculus courses. I used a mixed method approach to examine how students experience online calculus homework in order to provide insights as to how online homework might be improved. In particular, I examined the behaviors, perceptions, and resources associated with online homework. A survey was administered to all students in the mainstream calculus course that provides quantitative information about general trends and informs further questioning. For example, more than half of students reported that they never study calculus with classmates nor in office hours. In tandem with the large survey, I also closely studied the online homework experience of 4 students through screen recordings and interviews.



City Center B

StudentsÕ symmetric ability in relation to their use and preference for symmetry heuristics in problem solving

Meredith Muller and Eric Pandiscio

Advanced mathematical problem solving is marked by efficient and fluid use of multiple solution strategies. Symmetric arguments are apt heuristics and eminently useful in mathematics and science fields. Research suggests that mathematics proficiency is correlated with spatial reasoning. We define symmetric ability as fluency with mentally visualizing, manipulating, and making comparisons among 2D objects under rotation and reflection. We hypothesize that symmetric ability is a distinct sub-ability of spatial reasoning which is more accessible to students due to inherent cultural biases for symmetric balance. Do students with varying levels of symmetric ability use or prefer symmetric arguments in problem solving? How does symmetric ability relate to insight in problem solving? Results from a pilot study indicate that, among undergraduates, there is high variation in symmetric ability. Further, students with higher symmetric ability tend towards more positive attitudes about mathematics. Methods, future research, and implications are discussed.