Smith, Mountcastle, Thompson on the Boltzmann factor

T.I. Smith, D.B. Mountcastle, and J.R. Thompson

Student understanding of the Boltzmann factor

Phys. Rev. ST Phys. Educ. Res. 11, 020123 (2015).  

Published 23 September 2015.

[This paper is part of the Focused Collection on Upper Division Physics Courses.] We present results of our investigation into student understanding of the physical significance and utility of the Boltzmann factor in several simple models. We identify various justifications, both correct and incorrect, that students use when answering written questions that require application of the Boltzmann factor. Results from written data as well as teaching interviews suggest that many students can neither recognize situations in which the Boltzmann factor is applicable nor articulate the physical significance of the Boltzmann factor as an expression for multiplicity, a fundamental quantity of statistical mechanics. The specific student difficulties seen in the written data led us to develop a guided-inquiry tutorial activity, centered around the derivation of the Boltzmann factor, for use in undergraduate statistical mechanics courses. We report on the development process of our tutorial, including data from teaching interviews and classroom observations of student discussions about the Boltzmann factor and its derivation during the tutorial development process. This additional information informed modifications that improved students’ abilities to complete the tutorial during the allowed class time without sacrificing the effectiveness as we have measured it. These data also show an increase in students’ appreciation of the origin and significance of the Boltzmann factor during the student discussions. Our findings provide evidence that working in groups to better understand the physical origins of the canonical probability distribution helps students gain a better understanding of when the Boltzmann factor is applicable and how to use it appropriately in answering relevant questions.

Smith, Christensen, Mountcastle, and Thompson on entropy, heat engines, and the Carnot cycle

Trevor I. Smith, Warren M. Christensen, Donald B. Mountcastle, and John R. Thompson

Identifying student difficulties with entropy, heat engines, and the Carnot cycle

Phys. Rev. ST Phys. Educ. Res. 11, 020116 – Published 23 September 2015

[This paper is part of the Focused Collection on Upper Division Physics Courses.] We report on several specific student difficulties regarding the second law of thermodynamics in the context of heat engines within upper-division undergraduate 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.

Wittmann and Black on procedural resources in mathematics

Michael C. Wittmann and Katrina E. BLack

Mathematical actions as procedural resources: An example from the separation of variables.

Physical Review Special Topics - Physics Education Research, 11, 020114 - Published Sept 23, 2015

[This paper is part of the Focused Collection on Upper Division Physics Courses.] Students learning to separate variables in order to solve a differential equation have multiple ways of correctly doing so. The procedures involved in separation include division or multiplication after properly grouping terms in an equation, moving terms (again, at times grouped) from one location on the page to another, or simply carrying out separation as a single act without showing any steps. We describe student use of these procedures in terms of Hammer’s resources, showing that each of the previously listed procedures is its own “piece” of a larger problem solving activity. Our data come from group examinations of students separating variables while solving an air resistance problem in an intermediate mechanics class. Through detailed analysis of four groups of students, we motivate that the mathematical procedures are resources and show the issues that students must resolve in order to successfully separate variables. We use this analysis to suggest ways in which new resources (such as separation) come to be.