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Weyl&#39;s problem deals with the distribution of eigenvalues of the wave equation in a bounded domain. The integrated density of states counts the number of states up to a certain wavenumber and has important applications in nuclear physics, degenerate Fermi gases, blackbody radiation, and Bose-Einstein condensation. In the limit of large wavenumbers, the integrated density of states depends only on the volume of the domain and not on its shape. Corrections to this behaviour are well-known and depend on the surface area of the domain, its curvature and other features. We describe several computational projects that allow students to investigate this dependence for three different bounding domains – a rectangular box, a sphere and a circular cylinder. Quasi one- and two-dimensional systems can be analyzed by considering various limits. These projects could be incorporated into courses in quantum mechanics or statistical mechanics, or could stand alone in a computational physics course.

AAPT SM 20

Computational Physics Projects Related to Weyl’s Problem

weyl's problem

statistical mechanics

computational physics

quantum mechanics

The 2020 AAPT Virtual Summer Meeting brought together physics educators from around the globe to learn about the latest in physics theory, research methodology and education from leaders in their field. The program featured plenary speakers, sessions, workshops and many special activities.

For more information please visit the 2020 AAPT Virtual Summer Meeting page at: https://www.aapt.org/Conferences/SM2020/index.cfm.


The 2020 AAPT Virtual Summer Meeting brought together physics educators from around the globe to learn about the latest in physics theory, research methodology and education from leaders in their field. 

Weyl's problem deals with the distribution of eigenvalues of the wave equation in a bounded domain. The integrated density of states counts the number of states up to a certain wavenumber and has important applications in nuclear physics, degenerate Fermi gases, blackbody radiation, and Bose-Einstein condensation. In the limit of large wavenumbers, the integrated density of states depends only on the volume of the domain and not on its shape. Corrections to this behaviour are well-known and depend on the surface area of the domain, its curvature and other features. We describe several computational projects that allow students to investigate this dependence for three different bounding domains – a rectangular box, a sphere and a circular cylinder. Quasi one- and two-dimensional systems can be analyzed by considering various limits. These projects could be incorporated into courses in quantum mechanics or statistical mechanics, or could stand alone in a computational physics course.

technical paper

Students with disabilities make up between 10-20% of postsecondary students, however instructors lack knowledge about how to support this population and little research has investigated how courses can be designed to support students with disabilities. Universal Design for Learning (UDL) is one framework we can use to identify barriers to learning in our courses and identify strategies that reduce these barriers. UDL supports the reduction of barriers to learning by providing guidelines and checkpoints which provide considerations and recommendations for supporting the variability of students’ interests, needs, and abilities. We will present how the UDL framework was used in observations of postsecondary, introductory physics courses to identify barriers to learning. We will then exhibit inclusive practices that the instructors for the observed courses chose to implement in response to feedback from the observations, along with how students responded to the implemented practices.

Using Universal Design for Learning to Identify and Implement Inclusive Practices

The use of active-learning instructional strategies in high schools is strongly emphasized in the new Course of Study in Japan. However, in the actual classroom, one-way didactic teaching is more common. This study focuses on a high-school physics teacher and examines how his beliefs and teaching change through action research. The change of his teaching from the teacher-driven way before the treatment to more student-centered way after the treatment was shown by the use of S-T graphs and co-occurrence network diagrams of the observation data. It was interesting to notice that the teacher does not recognize the change in spite of the obvious change in his teaching style. Further discussions on the effects of action research for promoting active learning in high-school physics will be included in the presentation.

Improving High-School Physics Lessons through Action Research

Hypothesis–Experiment Class (HEC) is the educational approach proposed by Itakura in 1963. Since that time, a number of HEC curricular materials have been developed to teach students of all ages a wide range of topics in both natural and social sciences. Although Hatano and Inagaki introduced HEC to the West in the 1980s, it was done in service of discussing findings in cognitive science, and HEC itself has received little attention outside of Japan. That is beginning to change, however, with the recent translation and publication of the book “Hypothesis-Experiment Class (Kasetsu)” in 2019. HEC has been compared with the well-known Predict-Observe-Explain (POE) approach as well as other educational approaches, but there are some salient features that make HEC unique. In this presentation, we will discuss how HEC is similar to and different from other educational approaches, looking in particular at a specific HEC lesson on background radiation.

Rediscovery of a child-centered Japanese educational approach: Hypothesis–Experiment Class

Students' self-efficacy, their confidence in their ability to complete a task, is a good predictor for success and persistence. Previous research indicates that improvements in learning methods can improve student self-efficacy and that classroom dynamics may impact students' self-efficacy by allowing for different kinds of self-efficacy opportunities. In this study, we analyzed interviews from 12 students enrolled in a flipped integral calculus course to understand their perceptions of their self-efficacy and how this related to classroom dynamics and activities. Preliminary findings reveal that experiences in previous math courses, particularly high school, impacted students’ perceptions of their self-efficacy in math both positively and negatively, active learning activities increased students’ confidence in their ability to do math from their perspective, and verbal persuasion (implicit encouragement) increased students’ confidence and was seen as a helpful way to learn.

Qualitative analysis of students' perceptions of their self-efficacy

COR-400, Science and Human Nature, is a general education science course required of all students at Oglethorpe University. In this course, students learn about scientific ways of knowing and the scientific process through the lens of scientific revolutions. Specifically, the course content focuses on the Copernican Revolution and the Quantum Revolution of the early 20th century. Students often enter this course with preconceived notions of the relationship between religion and the scientific community – in particular, they often perceive religious authority to have produced barriers to scientific progress and discovery throughout history. During this course, we investigate and challenge these notions. In this presentation I will discuss how this material is presented to students and highlight student comments that demonstrate their evolution in thinking about the relationship between science and religion.

Challenging Student Ideas About Religion in a General Education Course

We have a large introductory physics course for non-majors, comprised mainly of First Year Undergraduate students. We provide for them a number of multiple choice and formula response questions online to help them prepare for the exams. While we do have other graded components of the course online, this part is set to be ungraded and allows them to see the solutions after a single attempt. We use data collected on the online student responses to explore student study behavior. For example, we were curious to see how students used this online answer checker to help them study for exams - did they try until they got the answer themselves or did they simply put in a junk answer in order to be able to read the solution?

How Students Use Ungraded Exam Review Problems

Social Comparison Concern (SCC) is a measurement of how negatively students compare themselves respect to their classmates based on their skill and knowledge of the class. Previous research found that female students, underrepresented minorities, and first generation students reported higher SCC on average, SCC was moderately (negatively) correlated with grade, and SCC mediated the effect of belonging on grade. To further understand SCC, we designed an online intervention by adapting prior motivational intervention designs, and administered a semester-long online SCC pilot intervention study for the first course in the introductory calculus physics sequence at the Ohio State University. Students were divided into SCC treatment or no-treatment control conditions by lecture section. We report on evidence that SCC may be impacting midterm and final exam scores (and vice-versa), especially for lower-scoring students, but the results of our pilot treatment remain inconclusive, and we will discuss our next steps.

The Impact of Grades on Social Comparison Concern

Physics education, especially astronomy education, is an ideal place to explore the intersection of science and religion. Grounded in tenets of peace education, the study of astronomy can be enriched by punctuating learning outcomes of physics concepts with the stories, mythlore, and teachings emanating from religion and culture. The purpose of this approach is to advance the physics we are teaching by resonating with students’ identities and being an entry to the greater conversation about (under)representation in physics, and how the intersection of religion and science provides insight into living and learning in a multi-religious and multicultural world. This session, which centers on the calendar by which cultural and religious observances are determined, advocates for integrating elements of cultural and religious knowledge with astronomy concepts as an inclusive space for all students. This pragmatic approach aligns with andragogical intentions of developing and delivering an increased holistic and inclusive curriculum.

A space oddity: Exploring the intersection of science and religion in physics classrooms.

[This abstract represents one part of a two-part study; some language is reproduced in another abstract.] Our team is studying student ownership of projects, a significant learning outcome in many project-based lab courses. An empirical model of ownership will support educators and researchers in the practice, design, and implementation of courses that encourage student ownership. Our preliminary model was based on analysis of a subset of survey and interview data from five Western and Midwestern physics departments. In this talk, I will demonstrate how this ownership model maps onto our full dataset, including stimulated recall interviews, and what educational implications emerge from further analysis. In prior work, we described ownership as a relationship between students and their projects. Material, cognitive, and affective student-project interactions composed the resulting ownership model. These interactions included student contributions to their projects, student understanding of project topics, and student emotional responses to progress and setbacks.

Student Ownership of Physics Projects: Manifestation in Student-Project Interactions

[This abstract represents one part of a two-part study; some language is reproduced in another abstract.] Our team is studying student ownership of projects, a significant learning outcome in many project-based lab courses. An empirical model of ownership will support educators and researchers in the practice, design, and implementation of courses that encourage student ownership. Our preliminary model was based on analysis of a subset of survey and interview data from five Western and Midwestern physics departments. In this talk, I will demonstrate how this ownership model maps onto our full dataset, including stimulated recall interviews, and what educational implications emerge from further analysis. In our model, ownership is more than an outcome that develops over time and is achieved by the project’s end. Students may or may not experience ownership in any of three phases: choice of topic, execution of methods, and synthesis of results into reports or presentations.

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Computational Physics Projects Related to Weyl’s Problem

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