In this study we investigated how two different approaches to drawing free body diagrams influence the development of students’ understanding of Newton’s laws, including their ability to identify real forces. For this purpose we developed a 12-item two-tier multiple choice survey and conducted a quasiexperiment. This experiment included two groups of first-year physics students from Rijeka (RG) ($n_e=27$) and Split (SG) ($n_c=25$) Universities. Students from both groups solved mechanics problems for a period of two class hours. The only difference was that RG students used the superposition of forces approach to solving mechanics problems and in SG the decomposition of forces approach has been used. The ANCOVA ($n_c=17$, $n_e=17$) showed a statistically significant difference in favor of RG, whereby the effect sizes were moderate to large, and the largest differences have been observed in the ability of identifying real forces. Students from the control group (SG) more often exhibited the misconception that forces and their components act on a body independently and simultaneously. Our results support the idea that the practice of resolving forces into the components may not be the most effective way to develop understanding of Newton’s laws and the concept of force.

This study was based on the hypothesis that students’ epistemological beliefs could become more expertlike with a combination of appropriate instructional activities: (i) preclass reading with metacognitive reflection, and (ii) in-class active learning that produces cognitive dissonance. This hypothesis was tested through a five-year study involving close to 1000 students at two institutions, in four physics courses. Using an experimental design, data from student interviews, writing product assessments, and the Discipline-Focused Epistemological Beliefs Questionnaire (DFEBQ) we demonstrate that the beliefs of novice science learners became more expertlike on 2 of the 4 DFEBQ factors. We conclude that a combination of an activity that gets students to examine textual material metacognitively (Reflective Writing) with one or more types of in-class active learning interventions can promote positive change in students’ epistemological beliefs.

Superposition gives rise to the probabilistic nature of quantum mechanics and is therefore one of the concepts at the heart of quantum mechanics. Although we have found that many students can successfully use the idea of superposition to calculate the probabilities of different measurement outcomes, they are often unable to identify the experimental implications of a superposition state. In particular, they fail to recognize how a superposition state and a mixed state (sometimes called a “lack of knowledge” state) can produce different experimental results. We present data that suggest that superposition in quantum mechanics is a difficult concept for students enrolled in sophomore-, junior-, and graduate-level quantum mechanics courses. We illustrate how an interactive lecture tutorial can improve student understanding of quantum mechanical superposition. A longitudinal study suggests that the impact persists after an additional quarter of quantum mechanics instruction that does not specifically address these ideas.

In this paper we present a series of item response models of data collected using the Force Concept Inventory. The Force Concept Inventory (FCI) was designed to poll the Newtonian conception of force viewed as a multidimensional concept, that is, as a complex of distinguishable conceptual dimensions. Several previous studies have developed single-trait item response models of FCI data; however, we feel that multidimensional models are also appropriate given the explicitly multidimensional design of the inventory. The models employed in the research reported here vary in both the number of fitting parameters and the number of underlying latent traits assumed. We calculate several model information statistics to ensure adequate model fit and to determine which of the models provides the optimal balance of information and parsimony. Our analysis indicates that all item response models tested, from the single-trait Rasch model through to a model with ten latent traits, satisfy the standard requirements of fit. However, analysis of model information criteria indicates that the five-trait model is optimal. We note that an earlier factor analysis of the same FCI data also led to a five-factor model. Furthermore the factors in our previous study and the traits identified in the current work match each other well. The optimal five-trait model assigns proficiency scores to all respondents for each of the five traits. We construct a correlation matrix between the proficiencies in each of these traits. This correlation matrix shows strong correlations between some proficiencies, and strong anticorrelations between others. We present an interpretation of this correlation matrix.

Broad instructional methods like “interactive engagement” have been shown to be effective, but such general characterization provides little guidance on the details of how to structure instructional materials. In this study, we seek instructional specificity by comparing two ways of using an analogy to learn a target physical principle: (i) applying the analogy to the target physical domain on a case-by-case basis and (ii) using the analogy to create a general rule in the target physical domain. In the discussion sections of a large, introductory physics course ($N=231$), students who sought a general rule were better able to discover and apply a correct physics principle than students who analyzed the examples case by case. The difference persisted at a reduced level after subsequent direct instruction. We argue that students who performed case-by-case analyses were more likely to focus on idiosyncratic problem-specific features rather than the deep structural features. This study provides an example of investigations into how the specific structure of instructional materials can be consequential for what is learned.

We report on seven years of attitudinal data using the Colorado Learning Attitudes about Science Survey from University Modeling Instruction (UMI) sections of introductory physics at Florida International University. University Modeling Instruction is a curricular and pedagogical transformation of introductory university physics that engages students in building and testing conceptual models in an integrated lab and lecture learning environment. This work expands upon previous studies that reported consistently positive attitude shifts in UMI courses; here, we disaggregate the data by gender and ethnicity to look for any disparities in the pattern of favorable shifts. We find that women and students from statistically underrepresented ethnic groups have gains that are comparable to those of men and students from well-represented ethnic groups on this attitudinal measure, and that this result holds even when interaction effects of gender and ethnicity are included. We conclude with suggestions for future work in UMI courses and for attitudinal equity investigations generally. We encourage researchers to expand their scope beyond simple performance gaps when considering equity concerns, and to avoid relying on a single measure to evaluate student success. Finally, we conjecture that students’ social and academic networks are one means by which attitudinal and efficacy beliefs about the course are propagated.

Separation of variables can be a powerful technique for solving many of the partial differential equations that arise in physics contexts. Upper-division physics students encounter this technique in multiple topical areas including electrostatics and quantum mechanics. To better understand the difficulties students encounter when utilizing the separation of variables technique, we examined students’ responses to midterm exam questions and a standardized conceptual assessment and conducted think-aloud, problem-solving interviews. Our analysis was guided by an analytical framework that focuses on how students activate, construct, execute, and reflect on the separation of variables technique when solving physics problems. Here we focus on student difficulties with separation of variables as a technique to solve Laplace’s equation in both Cartesian and spherical coordinates in the context of junior-level electrostatics. Challenges include recognizing when separation of variables is the appropriate tool, recalling or justifying the separated form of the potential and the need for the infinite sum, identifying implicit boundary conditions, and spontaneously reflecting on their solutions. Moreover, the type and frequency of errors was often different for separation of variables problems in Cartesian and spherical geometries. We also briefly discuss implication of these findings for instruction.

Standardized conceptual assessment represents a widely used tool for educational researchers interested in student learning within the standard undergraduate physics curriculum. For example, these assessments are often used to measure student learning across educational contexts and instructional strategies. However, to support the large-scale implementation often required for cross-institutional testing, it is necessary for these instruments to have question formats that facilitate easy grading. Previously, we created a multiple-response version of an existing, validated, upper-division electrostatics diagnostic with the goal of increasing the instrument’s potential for large-scale implementation. Here, we report on the validity and reliability of this new version as an independent instrument. These findings establish the validity of the multiple-response version as measured by multiple test statistics including item difficulty, item discrimination, and internal consistency. Moreover, we demonstrate that the majority of student responses to the new version are internally consistent even when they are incorrect and provide an example of how the new format can be used to gain insight into student difficulties with specific content in electrostatics.

Understanding Maxwell’s equations in differential form is of great importance when studying the electrodynamic phenomena discussed in advanced electromagnetism courses. It is therefore necessary that students master the use of vector calculus in physical situations. In this light we investigated the difficulties second year students at KU Leuven encounter with the divergence and curl of a vector field in mathematical and physical contexts. We have found that they are quite skilled at doing calculations, but struggle with interpreting graphical representations of vector fields and applying vector calculus to physical situations. We have found strong indications that traditional instruction is not sufficient for our students to fully understand the meaning and power of Maxwell’s equations in electrodynamics.

Students’ ability to effectively study for an exam, or to manage their time during an exam, is related to their metacognitive capacity. Prior research has demonstrated the effective use of metacognitive strategies during learning and retrieval is related to content expertise. Students also make judgments of their own learning and of problem difficulty to guide their studying. This study extends prior research by investigating the accuracy of novices’ and experts’ ability to judge problem difficulty across two experiments; here “accuracy” refers to whether or not their judgments of problem difficulty corresponds with actual exam performance in an introductory mechanics physics course. In the first experiment, physics education research (PER) experts judged the difficulty of introductory physics problems and provided the rationales behind their judgments. Findings indicate that experts use a number of different problem features to make predictions of problem difficulty. While experts are relatively accurate in judging problem difficulty, their content expertise may interfere with their ability to predict student performance on some question types. In the second experiment novices and “near experts” (graduate TAs) judged which question from a problem pair (taken from a real exam) was more difficult. The results indicate that judgments of problem difficulty are more accurate for those with greater content expertise, suggesting that the ability to predict problem difficulty is a trait of expertise which develops with experience.

Thermal science is challenging for students due to its largely imperceptible nature. Handheld infrared cameras offer a pedagogical opportunity for students to see otherwise invisible thermal phenomena. In the present study, a class of upper secondary technology students ($N=30$) partook in four IR-camera laboratory activities, designed around the predict-observe-explain approach of White and Gunstone. The activities involved central thermal concepts that focused on heat conduction and dissipative processes such as friction and collisions. Students’ interactions within each activity were videotaped and the analysis focuses on how a purposefully selected group of three students engaged with the exercises. As the basis for an interpretative study, a “thick” narrative description of the students’ epistemological and conceptual framing of the exercises and how they took advantage of the disciplinary affordance of IR cameras in the thermal domain is provided. Findings include that the students largely shared their conceptual framing of the four activities, but differed among themselves in their epistemological framing, for instance, in how far they found it relevant to digress from the laboratory instructions when inquiring into thermal phenomena. In conclusion, the study unveils the disciplinary affordances of infrared cameras, in the sense of their use in providing access to knowledge about macroscopic thermal science.

[This paper is part of the Focused Collection on Upper Division Physics Courses.] Partial derivatives are used in a variety of different ways within physics. Thermodynamics, in particular, uses partial derivatives in ways that students often find especially confusing. We are at the beginning of a study of the teaching of partial derivatives, with a goal of better aligning the teaching of multivariable calculus with the needs of students in STEM disciplines. In this paper, we report on an initial study of expert understanding of partial derivatives across three disciplines: physics, engineering, and mathematics. We report on the central research question of how disciplinary experts understand partial derivatives, and how their concept images of partial derivatives differ, with a focus on experimentally measured quantities. Using the partial derivative machine (PDM), we probed expert understanding of partial derivatives in an experimental context without a known functional form. In particular, we investigated which representations were cued by the experts’ interactions with the PDM. Whereas the physicists and engineers were quick to use measurements to find a numeric approximation for a derivative, the mathematicians repeatedly returned to speculation as to the functional form; although they were comfortable drawing qualitative conclusions about the system from measurements, they were reluctant to approximate the derivative through measurement. On a theoretical front, we found ways in which existing frameworks for the concept of derivative could be expanded to include numerical approximation.

[This paper is part of the Focused Collection on Upper Division Physics Courses.] Standardized assessment tests that allow researchers to compare the performance of students under various curricula are highly desirable. There are several research-based conceptual tests that serve as instruments to assess and identify students’ difficulties in lower-division courses. At the upper-division level assessing students’ difficulties is a more challenging task. Although several research groups are currently working on such tests, their reliability and validity are still under investigation. We analyze the results of the Colorado Upper-Division Electrostatics diagnostic from Oregon State University and compare it with data from the University of Colorado. In particular, we show potential shortcomings in the Oregon State University curriculum regarding separation of variables and boundary conditions, as well as uncover weaknesses of the rubric to the free-response version of the diagnostic. We also demonstrate how the diagnostic can be used to obtain information about student learning during a gap in instruction. Our work complements and extends the previous findings from the University of Colorado by highlighting important differences in student learning that may be related to the curriculum, illuminating difficulties with the rubric for certain problems and verifying decay in post-test results over time.

[This paper is part of the Focused Collection on Upper Division Physics Courses.] Most introductory quantum physics instructors would agree that transitioning students from classical to quantum thinking is an important learning goal, but may disagree on whether or how this can be accomplished. Although (and perhaps because) physicists have long debated the physical interpretation of quantum theory, many instructors choose to avoid emphasizing interpretive themes; or they discuss the views of scientists in their classrooms, but do not adequately attend to student interpretations. In this synthesis and extension of prior work, we demonstrate the following: (i) instructors vary in their approaches to teaching interpretive themes; (ii) different instructional approaches have differential impacts on student thinking; and (iii) when student interpretations go unattended, they often develop their own (sometimes scientifically undesirable) views. We introduce here a new modern physics curriculum that explicitly attends to student interpretations, and provide evidence-based arguments that doing so helps them to develop more consistent interpretations of quantum phenomena, more sophisticated views of uncertainty, and greater interest in quantum physics.

[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.

[This paper is part of the Focused Collection on Upper Division Physics Courses.] This study concentrates on evaluating the consistency of upper-division students’ use of the second law of thermodynamics at macroscopic and microscopic levels. Data were collected by means of a paper and pencil test ($N=48$) focusing on the macroscopic and microscopic features of the second law concerned with heat transfer processes. The data analysis was based on a qualitative content analysis where students’ responses to the macroscopic- and microscopic-level items were categorized to provide insight into the consistency of the students’ ideas; if students relied on the same idea at both levels, they ended up in the same category at both levels, and their use of the second law was consistent. The most essential finding is that a majority of students, 52%–69% depending on the physical system under evaluation, used the second law of thermodynamics consistently at macroscopic and microscopic levels; approximately 40% of the students used it correctly in terms of physics while others relied on erroneous ideas, such as the idea of conserving entropy. The most common inconsistency harbored by 10%–15% of the students (depending on the physical system under evaluation) was students’ tendency to consider the number of accessible microstates to remain constant even if the entropy was stated to increase in a similar process; other inconsistencies were only seen in the answers of a few students. In order to address the observed inconsistencies, we would suggest that lecturers should utilize tasks that challenge students to evaluate phenomena at macroscopic and microscopic levels concurrently and tasks that would guide students in their search for contradictions in their thinking.

[This paper is part of the Focused Collection on Upper Division Physics Courses.] Instructors and researchers think “thinking like a physicist” is important for students’ professional development. However, precise definitions and observational markers remain elusive. We reinterpret popular beliefs inventories in physics to indicate what physicists think thinking like a physicist entails. Through discourse analysis of upper-division students’ speech in natural settings, we show that students may appropriate or resist these elements. We identify a new element in the physicist speech genre: brief, embedded, spontaneous metacognitive talk (BESM talk). BESM talk communicates students’ in-the-moment enacted expectations about physics as a technical field and a cultural endeavor. Students use BESM talk to position themselves as physicists or nonphysicists. Students also use BESM talk to communicate their expectations in four ways: understanding, confusion, spotting inconsistencies, and generalized expectations.

[This paper is part of the Focused Collection on Upper Division Physics Courses.] As part of a longitudinal study into identity development in upper-level physics students, we used a phenomenographic research method to examine students’ perceptions of what it means to be a physicist. Analysis revealed six different categories of perception of what it means to be a physicist. We found the following themes: research and its association with being a physicist, differences in mindset, and exclusivity of accomplishments. The paper highlights how these perceptions relate to two communities of practice that the students are members of, and also highlights the importance of undergraduate research for students to transition from the physics undergraduate community of practice to the community of practicing physicists.

[This paper is part of the Focused Collection on Upper Division Physics Courses.] Compared with introductory physics, relatively little is known about the development of expertise in advanced physics courses, especially in the case of quantum mechanics. Here, we describe a framework for understanding the patterns of student reasoning difficulties and how students develop expertise in quantum mechanics. The framework posits that the challenges many students face in developing expertise in quantum mechanics are analogous to the challenges introductory students face in developing expertise in introductory classical mechanics. This framework incorporates both the effects of diversity in upper-level students’ prior preparation, goals, and motivation in general (i.e., the facts that even in upper-level courses, students may be inadequately prepared, have unclear goals, and have insufficient motivation to excel) as well as the “paradigm shift” from classical mechanics to quantum mechanics. The framework is based on empirical investigations demonstrating that the patterns of reasoning, problem-solving, and self-monitoring difficulties in quantum mechanics bear a striking resemblance to those found in introductory classical mechanics. Examples from research in quantum mechanics and introductory classical mechanics are discussed to illustrate how the patterns of difficulties are analogous as students learn to unpack the respective principles and grasp the formalism in each knowledge domain during the development of expertise. Embracing such a framework and contemplating the parallels between the difficulties in these two knowledge domains can enable researchers to leverage the extensive literature for introductory physics education research to guide the design of teaching and learning tools for helping students develop expertise in quantum mechanics.

[This paper is part of the Focused Collection on Upper Division Physics Courses.] As part of an ongoing project to examine student learning in upper-division courses in thermal and statistical physics, we have examined student reasoning about entropy and the second law of thermodynamics. We have examined reasoning in terms of heat transfer, entropy maximization, and statistical treatments of multiplicity and probability. In this paper, we describe student responses in interviews focused on the approach of macroscopic systems to thermal equilibrium. Our data suggest that students do not use a single simple model of entropy, but rather use a variety of conceptual resources. Individual students frequently shifted between resources, in some cases leading to contradictory predictions. Among the resources that students employed were some that have been previously described in the literature, including inappropriate use of conservation. However, our results suggest that student use of resources connected to disorder are neither simple nor monolithic. For example, many students used a previously unreported association between the equilibrium state of a system and an increase in order, rather than disorder.