10:15 – 10:45 am	Session 10 – Preliminary Reports
Marquis A	Does it converge? A look at second semester calculus students' struggles determining convergence of series David Earls and Eyob Demeke Despite the multitude of research that exists on student difficulty in first semester calculus courses, little is known about student difficulty determining convergence of sequences and series in second semester calculus courses. In our preliminary report, we attempt to address this gap specifically by analyzing student work from an exam question that asks students to determine the convergence of a series. We develop a framework that can be used to help analyze the mistakes students make when determining the convergence of series. In addition, we analyze how student errors relate to prerequisites they are expected to have entering the course, and how these errors are unique to knowledge about series. Paper 108
Marquis B	An example of a linguistic obstacle to proof construction: Dori and the hidden double negative Annie Selden and John Selden This paper considers the difficulty that university studentsÕ may have when unpacking an informally worded theorem statement into its formal equivalent in order to understand its logical structure, and hence, construct a proof. This situation is illustrated with the case of Dori who encountered just such a difficulty with a hidden double negative. She was taking a transition-to-proof course that began by having students first prove formally worded Òif-thenÓ theorem statements that enabled them to construct proof frameworks, and thereby, make initial progress on constructing proofs. But later, students were presented with some informally worded theorem statements to prove. We go on to consider the question of when, and how, to enculturate students into the often informal way that theorem statements are normally written, while still enabling them to progress in their proof construction abilities. Paper 63
Marquis C	Student characteristics and online retention: Preliminary investigation of factors relevant to mathematics course outcomes Claire Wladis, Alyse Hachey and Katherine Conway There is evidence that students drop out at higher rates from online than face-to-face courses, yet it is not well understood which students are particularly at risk online. In particular, online mathematics (and other STEM) courses have not been well-studied in the context of larger-scale analyses of online dropout. This study surveyed online and face-to-face students from a large U.S. university system. Results suggest that for online courses generally, grades are significant predictors of differential online versus face-to-face performance and that student parents may be particularly vulnerable to poor online course outcomes. Native-born students were also vulnerable online. The next stage of this research will be to analyze the factors that are relevant to online versus face-to-face retention in mathematics (and other STEM) courses specifically. Paper 69
Grand Ballroom 5	Student interpretation and justification of ÒbackwardÓ definite integrals Vicki Sealey and John Thompson The definite integral is an important concept in calculus, with applications throughout mathematics and science. Studies of student understanding of definite integrals reveal several student difficulties, some of which are related to determining the sign of an integral. Clinical interviews of 5 students gleaned their understanding of ÒbackwardÓ definite integrals, i.e., integrals for which the lower limit is greater than the upper limit and the differential is negative. Students initially invoked the Fundamental Theorem of Calculus to justify the negative sign. Some students eventually accessed the Riemann sum appropriately but could not determine how to obtain a negative quantity this way. We see the primary obstacle here as interpreting the differential as a width, and thus an unsigned quantity, rather than a difference between two values. Paper 80
City Center A	Divergent definitions of inquiry-based learning in undergraduate mathematics Samuel Cook, Sarah Murphy and Tim Fukawa-Connelly Inquiry-based learning is becoming more important and widely practiced in undergraduate mathematics education. As a result, research about inquiry-based learning is similarly becoming more common, including questions of the efficacy of such methods. Yet, thus far, there has been little effort on the part of practitioners or researchers to come to a description of the range(s) of practice that can or should be understood as inquiry-based learning. As a result, studies, comparisons and critiques can be dismissed as not using the appropriate definition, without adjudicating the quality of the evidence or implications for research and teaching. Through a large-scale literature review and surveying of experts in the community, this study begins the conversation about possible areas of agreement that would allow for a constituent definition of inquiry-based learning and allow for differentiation with non-inquiry pedagogical practices. Paper 88
City Center B	IVT as a starting point for multiple real analysis topics Steve Strand The proof of the Intermediate Value Theorem (IVT) provides a rich and approachable context for motivating many concepts central to real analysis, such as: sequence and function convergence, completeness of the real numbers, and continuity. As a part of the development of local instructional theory, an RME-based design experiment was conducted in which two post-calculus undergraduate students developed techniques to approximate the root of a polynomial. They then adapted those techniques into a (rough) proof of the IVT. Paper 118

Friday