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9:15 – 9:45 am

Session 20 – Preliminary Reports

Marquis A

Calculus students' understanding of logical implication and its relationship to their understanding of calculus theorems

Joshua Case and Natasha Speer

In undergraduate mathematics, deductive reasoning is an important skill for learning theoretical ideas and is primarily characterized by the concept of logical implication. This plays roles whenever theorems are applied, i.e., one must first check if a theoremÕs hypotheses are satisfied and then make correct inferences. In calculus, students must learn how to apply theorems. However, most undergraduates have not received instruction in propositional logic. How do these students comprehend the abstract notion of logical implication and how do they reason conditionally with calculus theorems? Results from our study indicated that students struggled with notions of logical implication in abstract contexts, but performed better when working in calculus contexts. Strategies students used (successfully and unsuccessfully) were characterized. Findings indicate that some students use Ņexample generatingÓ strategies to successfully determine the validity of calculus implications. Background on current literature, results of our study, further avenues of inquiry, and instructional implications are discussed.



Marquis B

Service-learning in a precalculus class: Tutoring improves the course performance of the tutor.

Ekaterina Yurasovskaya

We have introduced an experiment: as part of a Precalculus class, university students have been tutoring algebra prerequisites to students from the community via an academic service-learning program. The goal of the experiment was to improve university studentsÕ mastery of basic algebra and to quantitatively describe benefits of service-learning to studentsÕ performance in mathematics. At the end of the experiment, we observed 59% decrease of basic algebraic errors between experimental and control sections. The setup and analysis of the study have been informed by the theoretical research on service-learning and peer learning, both grounded in the constructivist theory of John Dewey.



Marquis C

How well prepared are preservice elementary teachers to teach early algebra?

Funda Gonulates, Leslie Nabors Olah, Heejoo Suh, Xueying Ji and Heather Howell

This study aims to investigate undergraduate preservice teachersÕ knowledge in the domain of early algebra. We conducted 90-minute clinical interviews with 15 preservice teachers in their fourth year of a five-year teacher preparation program. These interview sessions collected preservice teachersÕ responses to a series of assessment items designed to measure their content knowledge for teaching early algebra, with follow up questions probing their content-based reasoning. We also collected self-report of their preparation in this content area. We found that the participants had difficulty on the items targeting the meaning and use of operational properties and also in evaluating the appropriate use of the equal sign when presented with different uses in student work. They reported that they had few opportunities to learn about early algebra as mathematical content and as a topic to teach. These findings can inform the development of teacher-education curricula and support materials.



Grand Ballroom 5

Undergraduate students proof-reading strategies: A case study at one research institution

Eyob Demeke and Mateusz Pacha-Sucharzewski

Weber (2015) identified five effective proof-reading strategies that undergraduate students in proof-based courses can use to facilitate their proof comprehension. Following WeberÕs (2015) study, we designed a survey study to examine how undergraduate studentsÕ proof-reading strategies relate to what proficient learners of mathematics (mathematics professors and graduate students in mathematics) say undergraduates should employ when reading proofs. Our preliminary findings are: (i) Majority of the professors in our study claimed that undergraduates should use the strategies identified in WeberÕs (2015) study, (ii) ProfessorsÕ response significantly differed from undergraduatesÕ in only two of the five proof-reading strategies described in WeberÕs (2015) study (attempting to prove theorem before reading its proof and illustrating confusing assertions with examples), and finally (iii) Undergraduate students, for the most part, tend to agree with their professorsÕ preferred proof-reading strategies.



City Center A

Beyond procedures: Quantitative reasoning in upper-division Math Methods in Physics

Michael Loverude

Many upper-division physics courses have as goals that students should Ōthink like a physicist.Õ While this is not well-defined, most would agree that thinking like a physicist includes quantitative reasoning skills: considering limiting cases, dimensional analysis, and using approximations. However, there is often relatively little curricular support for these practices and many instructors do not assess them explicitly. As part of a project to investigate student learning in math methods, we have developed a number of written questions testing the extent to which students in an upper-division course in Mathematical Methods in Physics can employ these skills. Although there are limitations to assessing these skills with written questions, they can provide insight to the extent to which students can apply a given skill when prompted.



City Center B

A case for whole class discussions: Two case studies of the interaction between instructor role and instructor experience with a research-informed curriculum

Aaron Wangberg, Elizabeth Gire, Brian Fisher and Jason Samuels

This paper presents case studies of two instructors implementing a research informed multivariable calculus curriculum. The analysis, structured around social constructivist concepts, focuses on the interactions between the roles of the instructor in facilitating student discussions and the instructorsÕ experiences with the activities. This study is a part of an effort to evaluate and improve the projectÕs effectiveness in supporting instructors in implementing the activities to promote rich discussions with and among students. We find these instructors to be focused on their roles as facilitators for student-centered small-group discussion and that they choose not to have of whole class discussions. We argue that initiating whole class discussions would address concerns and negative experiences reported by the instructors.