3:25 – 3:55 pm	Session 16 – Preliminary Reports
Marquis A	Effect of teacher prompts on student proof construction Margaret Morrow and Mary Shepherd Many students have difficulty learning to construct mathematical proofs. In an upper level mathematics course using inquiry based methods, while this is some research on the types of verbal discourse in these courses, there is little, if any, research on teachersÕ written comments on studentsÕ work. This paper presents some very preliminary results from ongoing analysis from one teacherÕs written prompts on studentsÕ rough drafts of proofs for an Abstract Algebra course. The teacher prompts will initially be analyzed through a framework proposed by Blanton & Stylianou (2014) for verbal discourse and the framework will be modified in the course of the analysis. Can we understand if a type of prompt is ÒbetterÓ in some sense in getting students to reflect on their work and refine their proofs? It is anticipated that teacher prompts in the form of transactive questions are more effective in helping students construct proofs. Paper 64
Marquis B	Secondary teachers confronting mathematical uncertainty: Reactions to a teacher assessment item on exponents Heejoo Suh, Heather Howell and Yvonne Lai Teaching is inherently uncertain, and teaching secondary mathematics is no exception. We take the view that uncertainty can present opportunity for teachers to refine their practice, and that undergraduate mathematical preparation for secondary teaching can benefit from engaging pre-service teachers in tasks presenting uncertainty. We examined 13 secondary teachersÕ reactions to mathematical uncertainty when engaged with concepts about extending the domain for the operation of exponentiation. The data are drawn from an interview-based study of items developed to assess mathematical knowledge for teaching at the secondary level. In our findings, we characterize ways in which teachers either denied or mathematically investigated the uncertainty. Potential implications for instructors include using mathematical uncertainty to provide an opportunity for undergraduates to learn both content and practices of the Common Core State Standards. The proposal concludes with questions addressing how undergraduate mathematics instructors could use uncertainty as a resource when teaching preservice teachers. Paper 70
Marquis C	StudentsÕ concept image of tangent line compared to their understanding of the definition of the derivative Brittany Vincent and Vicki Sealey Our research explores first-semester calculus students' understanding of tangent lines and the derivative concept through a series of three interviews conducted over the course of one semester. Using a combination of Zandieh's (2000) derivative framework and Tall and Vinner's (1981) notions of concept image and concept definition, our analysis examines the role that students' concept image of tangent lines plays in their conceptual understanding of the derivative concept. Preliminary results seem to indicate that students are more successful when their concept image of tangent includes the limiting position of secant lines, as opposed to a tangent line as the line that touches the curve at one point. Paper 89
Grand Ballroom 5	Unraveling, synthesizing and reweaving: Approaches to constructing general statements. Duane Graysay Learning progressions for the development of the ability to look for and make use of mathematical structure would benefit from understanding how students in mathematics-focused majors might construct such structures in the form of general statements. The author recruited ten university students to interviews focused on tasks that asked for the reconstruction of a general statement to accommodate a broader domain. Through comparative analysis of responses, four major categories of approaches to such tasks were identified. This preliminary report describes in brief those four categories. Paper 102
City Center A	Student conceptions of definite integration and accumulation functions Brian Fisher, Jason Samuels and Aaron Wangberg Prior research has shown several common student conceptualizations of integration among undergraduates. This report focuses on data from a written assessment of studentsÕ views on definite integration and accumulation functions to categorize student conceptualizations and report on their prevalence among the undergraduate population. Analysis of these results found four categorizations for student descriptions of definite integrals: antiderivative, area, an infinite sum of one dimensional pieces, and a limit of approximations. When asked about an accumulation function, student responses were grouped into three categorizations: those based on the process of calculating a single definite integral, those based on the result of calculating a definite integral, and those based on the relationship between changes in the input and output variables of the accumulation function. These results were collected as part of a larger study on student learning in multivariable calculus, and the implications of these results on multivariable calculus will be considered. Paper 113
City Center B	Supporting students in seeing sequence convergence in Taylor series convergence Jason Martin, Matthew Thomas and Michael Oehrtman Virtual manipulatives designed to increase student understanding of the concepts of approximation by Taylor polynomials and convergence of Taylor series were used in calculus courses at multiple institutions. 225 students responded to tasks requiring graphing Taylor polynomials, graphing Taylor series, and describing relationships between different notions of convergence. We detail significant differences observed between students who used virtual manipulatives and those that did not. We propose that the use of these virtual manipulatives promotes an understanding of Taylor series supporting an understanding consistent with the formal definition of pointwise convergence. Paper 115

Friday