Friday

11:35 – 12:05 pm

Session 12 – Mixed Reports

Marquis A

Results from a national survey of abstract algebra instructors: Math ed is solving problems they don't have

Tim Fukawa-Connelly, Estrella Johnson and Rachel Keller

There is significant interest from policy boards and funding agencies to change studentsŐ experiences in undergraduate mathematics classes. Abstract algebra specifically has been the subject of reform initiatives, including new curricula and pedagogies, since at least the 1960s; yet there is little evidence about whether these change initiatives have proven successful. Pursuant to answering this question, we conducted a survey of abstract algebra instructors to generally investigate typical practices, and more specifically, their knowledge, goals, and orientations towards teaching and learning. On average, moderate levels of satisfaction were reported with regard to the course itself or student outcomes; moreover, little interest in, or knowledge of, reform practices or curricula were identified. We found that 77% of respondents spend the majority of class time lecturing – not surprising when considering 82% reported the belief that lecture is the most effective way to teach.

Paper

122

Marquis B

Investigating a mathematics graduate studentŐs construction of a hypothetical learning trajectory

Ashley Duncan

This study reports results of how a teacherŐs mathematical meanings and instructional planning decisions transformed while participating in and then generating a hypothetical learning trajectory on angles, angle measure and the radius as a unit of measurement. Using a teaching experiment methodology, an initial clinical interview was designed to reveal the teacherŐs meanings for angles and angle measure and to gain information about the teacherŐs instructional planning decisions. The teacher participated in a researcher generated HLT designed to promote the construction of productive meanings for angles and angle measure and then constructed her own HLT for her students. The initial interview revealed that the teacher had several unproductive meanings for angles and angle measure that caused the teacher perturbations while participating in the tasks of the researcher generated HLT. This participation allowed her to construct different meanings for angles and angle measure which changed her instructional planning decisions.

Paper

117

Marquis C

On the axiomatic formalization of mathematical understanding

Daniel Cheshire

This study adopts a property-based perspective to investigate the forms of abstraction, instantiation, and representation used by undergraduate topology students when acting to understand and use the concept of a continuous function as it is defined axiomatically. Based on a series of task-based interviews, profile cases are being developed to compare and contrast the distinct ways of thinking and processes of understanding observed by students undergoing this transition. A framework has been established to interpret the participantsŐ interactions with the underlying mathematical properties of continuous functions while they reconstructed their concept images to reflect a topological (axiomatic) structure. This will provide insight into how such properties can be successfully incorporated into studentsŐ concept images and accessed; and which obstacles prevent this. Preliminary results reveal several coherent categories of participantsŐ progression of understanding. This report will outline these profiles and seek critical feedback on the direction of the described research.

Paper

101

Grand Ballroom 5

Struggling to comprehend the zero-product property

John Paul Cook

The zero-product property (ZPP), typically stated as Ôif ab=0 then a=0 or b=0,Ő is an important property in school algebra (as a technique for solving equations) and abstract algebra (as the defining characteristic of integral domains). While the struggles of secondary mathematics students to employ the ZPP are well-documented, it unclear how undergraduate students preparing to take abstract algebra understand the ZPP as they enter abstract algebra. To this end, this paper documents studentsŐ understanding of the ZPP while also investigating how students might be able to develop and harness their own intuitive understandings of the property. Preliminary findings, in addition to characterizing how students reason with and understand the ZPP, indicate that the ZPP is no less trivial for advanced undergraduate students than it is for secondary mathematics student⹳

Paper

102