We suggest one redefinition of common clusters of questions used to analyze student responses on the Force and Motion Conceptual Evaluation. Our goal is to propose a methodology that moves beyond an analysis of student learning defined by correct responses, either on the overall test or on clusters of questions defined solely by content. We use the resources framework theory of learning to define clusters within this experimental test that was designed without the resources framework in mind. We take special note of the contextual and representational dependence of questions with seemingly similar physics content. We analyze clusters in ways that allow the most common incorrect answers to give as much, or more, information as the correctness of responses in that cluster. We show that false positives can be found, especially on questions dealing with Newton's third law. We apply our clustering to a small set of data to illustrate the value of comparing students' incorrect responses which are otherwise identical on a correct or incorrect analysis. Our work provides a connection between theory and experiment in the area of survey design and the resources framework.

One goal of physics instruction is to have students learn to make physical meaning of specific mathematical ideas, concepts, and procedures in different physical settings. As part of research investigating student learning in statistical physics, we are developing curriculum materials that guide students through a derivation of the Boltzmann factor, using a Taylor series expansion of entropy. Using results from written surveys, classroom observations, and both individual thinkaloud and teaching interviews, we present evidence that many students can recognize and interpret series expansions, but they often lack fluency with the Taylor series despite previous exposures in both calculus and physics courses. We present students' successes and failures both using and interpreting Taylor series expansions in a variety of contexts.

Previously, we analyzed the Force and Motion Conceptual Evaluation in terms of a resources-based model that allows for clustering of questions so as to provide useful information on how students correctly or incorrectly reason about physics. In this paper, we apply model analysis to show that the associated model plots provide more information regarding the results of investigations using these question clusters than normalized gain graphs. We provide examples from two different institutions to show how the use of model analysis with our redefined clusters can provide previously hidden insight into the effectiveness of instruction.

We report on several specific student difficulties regarding the Second Law of Thermodynamics in the context of heat engines within upper-division undergraduates thermal physics courses. Data come from ungraded written surveys, graded homework assignments, and videotaped classroom observations of tutorial activities. Written data show that students in these courses do not clearly articulate the connection between the Carnot cycle and the Second Law after lecture instruction. This result is consistent both within and across student populations. Observation data provide evidence for myriad difficulties related to entropy and heat engines, including students' struggles in reasoning about situations that are physically impossible and failures to differentiate between differential and net changes of state properties of a system. Results herein may be seen as the application of previously documented difficulties in the context of heat engines, but others are novel and emphasize the subtle and complex nature of cyclic processes and heat engines, which are central to the teaching and learning of thermodynamics and its applications. Moreover, the sophistication of these difficulties is indicative of the more advanced thinking required of students at the upper division, whose developing knowledge and understanding give rise to questions and struggles that are inaccessible to novices.

Mathematical reasoning skills are a desired outcome of many introductory physics courses, particularly calculus-based physics courses. Positive and negative quantities are ubiquitous in physics, and the sign carries important and varied meanings. Novices can struggle to understand the many roles signed numbers play in physics contexts, and recent evidence shows that unresolved struggle can carry over to subsequent physics courses. The mathematics education research literature documents the cognitive challenge of conceptualizing negative numbers as mathematical objects-both for experts, historically, and for novices as they learn. We contribute to the small but growing body of research in physics contexts that examines student reasoning about signed quantities and reasoning about the use and interpretation of signs in mathematical models. In this paper we present a framework for categorizing various meanings and interpretations of the negative sign in physics contexts, inspired by established work in algebra contexts from the mathematics education research community. Such a framework can support innovation that can catalyze deeper mathematical conceptualizations of signed quantities in the introductory courses and beyond.