[This paper is part of the Focused Collection on Upper Division Physics Courses.] Learning advanced physics, in general, is challenging not only due to the increased mathematical sophistication but also because one must continue to build on all of the prior knowledge acquired at the introductory and intermediate levels. In addition, learning quantum mechanics can be especially challenging because the paradigms of classical mechanics and quantum mechanics are very different. Here, we review research on student reasoning difficulties in learning upper-level quantum mechanics and research on students’ problem-solving and metacognitive skills in these courses. Some of these studies were multiuniversity investigations. The investigations suggest that there is large diversity in student performance in upper-level quantum mechanics regardless of the university, textbook, or instructor, and many students in these courses have not acquired a functional understanding of the fundamental concepts. The nature of reasoning difficulties in learning quantum mechanics is analogous to reasoning difficulties found via research in introductory physics courses. The reasoning difficulties were often due to overgeneralizations of concepts learned in one context to another context where they are not directly applicable. Reasoning difficulties in distinguishing between closely related concepts and in making sense of the formalism of quantum mechanics were common. We conclude with a brief summary of the research-based approaches that take advantage of research on student difficulties in order to improve teaching and learning of quantum mechanics.

[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.

[This paper is part of the Focused Collection on Upper Division Physics Courses.] The use of validated conceptual assessments alongside conventional course exams to measure student learning in introductory courses has become standard practice in many physics departments. These assessments provide a more standard measure of certain learning goals, allowing for comparisons of student learning across instructors, semesters, institutions, and pedagogies. Researchers at the University of Colorado Boulder have developed several similar assessments designed to target the more advanced physics of upper-division classical mechanics, electrostatics, quantum mechanics, and electrodynamics courses. Here, we synthesize the existing research on our upper-division assessments and discuss some of the barriers and challenges associated with their development, validation, and implementation as well as some of the strategies we have used to overcome these barriers.

[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.

[This paper is part of the Focused Collection on Upper Division Physics Courses.] We review and extend existing frameworks on modeling to develop a new framework that describes model-based reasoning in introductory and upper-division physics laboratories. Constructing and using models are core scientific practices that have gained significant attention within K–12 and higher education. Although modeling is a broadly applicable process, within physics education, it has been preferentially applied to the iterative development of broadly applicable principles (e.g., Newton’s laws of motion in introductory mechanics). A significant feature of the new framework is that measurement tools (in addition to the physical system being studied) are subjected to the process of modeling. Think-aloud interviews were used to refine the framework and demonstrate its utility by documenting examples of model-based reasoning in the laboratory. When applied to the think-aloud interviews, the framework captures and differentiates students’ model-based reasoning and helps identify areas of future research. The interviews showed how students productively applied similar facets of modeling to the physical system and measurement tools: construction, prediction, interpretation of data, identification of model limitations, and revision. Finally, we document students’ challenges in explicitly articulating assumptions when constructing models of experimental systems and further challenges in model construction due to students’ insufficient prior conceptual understanding. A modeling perspective reframes many of the seemingly arbitrary technical details of measurement tools and apparatus as an opportunity for authentic and engaging scientific sense making.

[This paper is part of the Focused Collection on Upper Division Physics Courses.] The time evolution of quantum states is arguably one of the more difficult ideas in quantum mechanics. In this article, we report on results from an investigation of student understanding of this topic after lecture instruction. We demonstrate specific problems that students have in applying time dependence to quantum systems and in recognizing the key role of the energy eigenbasis in determining the time dependence of wave functions. Through analysis of student responses to a set of four interrelated tasks, we categorize some of the difficulties that underlie common errors. The conceptual and reasoning difficulties that have been identified are illustrated through student responses to four sets of questions administered at different points in a junior-level course on quantum mechanics. Evidence is also given that the problems persist throughout undergraduate instruction and into the graduate level.

[This paper is part of the Focused Collection on Upper Division Physics Courses.] Energy measurements play a fundamental role in the theory of quantum mechanics, yet there is evidence that the underlying concepts are difficult for many students, even after all undergraduate instruction. We present results from an investigation into student ability to determine the possible energies that can be measured for a given wave function and Hamiltonian, to determine the probabilities of each energy measurement and how they depend on time, and to recognize how a measurement of energy affects the state. By analyzing student responses to open-ended questions, we identify five broad, interrelated sets of conceptual and reasoning difficulties related to energy measurements. Data are drawn from sophomore-, junior-, and graduate-level quantum mechanics courses. Particular attention is paid to incorrect ideas that persist across all levels.

[This paper is part of the Focused Collection on Upper Division Physics Courses.] In response to the need for a scalable, institutionally supported model of educational change, the Science Education Initiative (SEI) was created as an experiment in transforming course materials and faculty practices at two institutions—University of Colorado Boulder (CU) and University of British Columbia. We find that this departmentally focused model of change, which includes an explicit focus on course transformation as supported by a discipline-based postdoctoral education specialist, was generally effective in impacting courses and faculty across the institution. In CU’s Department of Physics, the SEI effort focused primarily on upper-division courses, creating high-quality course materials, approaches, and assessments, and demonstrating an impact on student learning. We argue that the SEI implementation in the CU Physics Department, as compared to that in other departments, achieved more extensive impacts on specific course materials, and high-quality assessments, due to guidance by the physics education research group—but with more limited impact on the departmental faculty as a whole. We review the process and progress of the SEI Physics at CU and reflect on lessons learned in the CU Physics Department in particular. These results are useful in considering both institutional and faculty-led models of change and course transformation.

[This paper is part of the Focused Collection on Upper Division Physics Courses.] The formalism of quantum mechanics includes a rich collection of representations for describing quantum systems, including functions, graphs, matrices, histograms of probabilities, and Dirac notation. The varied features of these representations affect how computations are performed. For example, identifying probabilities of measurement outcomes for a state described in Dirac notation may involve identifying expansion coefficients by inspection, but if the state is described as a function, identifying those expansion coefficients often involves performing integrals. In this study, we focus on three notational systems: Dirac notation, algebraic wave-function notation, and matrix notation. These quantum notations must include information about basis states and their associated complex probability amplitudes. In this theory paper, we identify four structural features of quantum notations, which we term individuation, degree of externalization, compactness, and symbolic support for computational rules. We illustrate how student reasoning interacts with these structural features with episodes from interviews with advanced undergraduate physics majors reasoning about a superposition state of an infinite square well system. We find evidence of the students coordinating different notations through the use of Dirac notation, using an expression in Dirac notation to guide their work in another notation. These uses are supported by the high degree of individuation, compactness, and symbolic support for computation and the moderate degree of externalization provided by Dirac notation.

[This paper is part of the Focused Collection on Upper Division Physics Courses.] We present a study of active learning pedagogies in an upper-division physics course. This work was guided by the principle of deliberate practice for the development of expertise, and this principle was used in the design of the materials and the orchestration of the classroom activities of the students. We present our process for efficiently converting a traditional lecture course based on instructor notes into activities for such a course with active learning methods. Ninety percent of the same material was covered and scores on common exam problems showed a 15% improvement with an effect size greater than 1 after the transformation. We observe that the improvement and the associated effect size is sustained after handing off the materials to a second instructor. Because the improvement on exam questions was independent of specific problem topics and because the material tested was so mathematically advanced and broad (including linear algebra, Fourier transforms, partial differential equations, and vector calculus), we expect the transformation process could be applied to most upper-division physics courses having a similar mathematical base.

Women currently outnumber men in obtaining undergraduate degrees but are underrepresented within STEM fields. However, women’s representation varies by STEM field, and even further by STEM subfield. One field that has held a persistent low representation of women is physics. This paper seeks to uncover the truth behind an anecdotal claim that the subfield of physics education research (PER) has a higher representation of women than physics as a whole. Graduate students in PER completed an online survey to assess their demographics, trajectory in PER, climate experiences, and goals for their research. The response rate for the survey was 68%, yielding 125 total respondents. This paper will focus on the 91 respondents enrolled in U.S. graduate programs. It was found that women make up 51% of the U.S. PER graduate students in this sample, as compared to only 19% of physics graduate students overall. Survey findings also revealed that both women and men in PER graduate programs experience similarly positive working relationships with faculty and fellow students. Last, both men and women reported building a stronger scientific workforce and becoming better teachers as goals for their PER research.

Problem solving is a critical element of learning physics. However, traditional instruction often emphasizes the quantitative aspects of problem solving such as equations and mathematical procedures rather than qualitative analysis for selecting appropriate concepts and principles. This study describes the development and evaluation of an instructional approach called Conceptual Problem Solving (CPS) which guides students to identify principles, justify their use, and plan their solution in writing before solving a problem. The CPS approach was implemented by high school physics teachers at three schools for major theorems and conservation laws in mechanics and CPS-taught classes were compared to control classes taught using traditional problem solving methods. Information about the teachers’ implementation of the approach was gathered from classroom observations and interviews, and the effectiveness of the approach was evaluated from a series of written assessments. Results indicated that teachers found CPS easy to integrate into their curricula, students engaged in classroom discussions and produced problem solutions of a higher quality than before, and students scored higher on conceptual and problem solving measures.

It is well known that introductory physics students often have alternative conceptions that are inconsistent with established physical principles and concepts. Invoking alternative conceptions in the quantitative problem-solving process can derail the entire process. In order to help students solve quantitative problems involving strong alternative conceptions correctly, appropriate scaffolding support can be helpful. The goal of this study is to examine how different scaffolding supports involving analogical problem-solving influence introductory physics students’ performance on a target quantitative problem in a situation where many students’ solution process is derailed due to alternative conceptions. Three different scaffolding supports were designed and implemented in calculus-based and algebra-based introductory physics courses involving 410 students to evaluate the level of scaffolding needed to help students learn from an analogical problem that is similar in the underlying principles involved but for which the problem-solving process is not derailed by alternative conceptions. We found that for the quantitative problem involving strong alternative conceptions, simply guiding students to work through the solution of the analogical problem first was not enough to help most students discern the similarity between the two problems. However, if additional scaffolding supports that directly helped students examine and repair their knowledge elements involving alternative conceptions were provided, e.g., by guiding students to contemplate related issues and asking them to solve the targeted problem on their own first before learning from the analogical problem provided, students were more likely to discern the underlying similarities between the problems and avoid getting derailed by alternative conceptions when solving the targeted problem. We also found that some scaffolding supports were more effective in the calculus-based course than in the algebra-based course. This finding emphasizes the fact that appropriate scaffolding support which is commensurate with students’ prior knowledge and skills must be determined via research in order to be effective.

It is known that experts identify or perceive meaningful patterns in visual stimuli related to their domain of expertise. This study explores the speed with which experts and novices detect changes in physics diagrams. Since change detection depends on where individuals direct their attention, differences in the speed with which experts and novices detect changes to diagrams would suggest differences in attention allocation between experts and novices. We present data from an experiment using the “flicker technique,” in which both physics experts and physics novices viewed nearly identical pairs of diagrams that are representative of typical introductory physics situations. The two diagrams in each pair contain a subtle difference that either does or does not change the underlying physics depicted in the diagram. Findings indicate that experts are faster at detecting physics-relevant changes than physics-irrelevant changes; however, there is no difference in response time for novices, suggesting that expertise guides attention for experts when inspecting physics diagrams. We discuss the cognitive implications of our findings.

We have conducted an investigation into how well students in introductory science classes (both physics and chemistry) are able to predict which questions they will or will not be able to answer correctly on an upcoming assessment. An examination of the data at the level of students’ overall scores reveals results consistent with the Dunning-Kruger effect, in which low-performing students tend to overestimate their abilities, while high-performing students estimate their abilities more accurately. Similar results have been widely reported in the science education literature. Breaking results out by students’ responses to individual questions, however, reveals that students of all ability levels have difficulty distinguishing questions which they are able to answer correctly from those that they are not able to answer correctly. These results have implications for the future study and reporting of students’ metacognitive abilities.

In this paper, we report about the development and validation of a learning progression about the Celestial Motion big idea. Existing curricula, research studies on alternative conceptions about these phenomena, and students’ answers to an open questionnaire were the starting point to develop initial learning progressions about change of seasons, solar and lunar eclipses, and Moon phases; then, a two-tier multiple choice questionnaire was designed to validate and improve them. The questionnaire was submitted to about 300 secondary students of different school levels (14 to 18 years old). Item response analysis and curve integral method were used to revise the hypothesized learning progressions. Findings support that spatial reasoning is a key cognitive factor for building an explanatory framework for the Celestial Motion big idea, but also suggest that causal reasoning based on physics mechanisms underlying the phenomena, as light flux laws or energy transfers, may significantly impact a students’ understanding. As an implication of the study, we propose that the teaching of the three discussed astronomy phenomena should follow a single teaching-learning path along the following sequence: (i) emphasize from the beginning the geometrical aspects of the Sun-Moon-Earth system motion; (ii) clarify consequences of the motion of the Sun-Moon-Earth system, as the changing solar radiation flow on the surface of Earth during the revolution around the Sun; (iii) help students moving between different reference systems (Earth and space observer’s perspective) to understand how Earth’s rotation and revolution can change the appearance of the Sun and Moon. Instructional and methodological implications are also briefly discussed.

Many topics in introductory astronomy at the college or high-school level rely implicitly on using astronomical photographs and visual data in class. However, students bring many preconceptions to their understanding of these materials that ultimately lead to misconceptions, and research about students’ interpretation of astronomical images has been scarcely conducted. In this study we probed college students’ understanding of astronomical photographs and visual data about galaxies and spectra, and developed a set of concept questions based on their common misconceptions. The study was conducted mainly in three successive surveys: (i) open-ended questions looking for students’ ideas and common misconceptions, (ii) combined multiple-choice and open-ended questions seeking to explore student reasoning and to improve concept questions for clickers, and (iii) a finalized version of the concept questions used to investigate the strength of each misconception among the students in introductory astronomy courses. This study reports on the procedures and the development of the concept questions with the investigated common misconceptions about galaxies and spectra. We also provide the set of developed questions for teachers and instructors seeking to implement in their classes for the purpose of formative assessment with the use of classroom response systems. These questions would help them recognize the gap between their teaching and students’ understanding, and ultimately improve teaching of the concepts.