We developed and validated a conceptual survey that focuses on the formalism and postulates of quantum mechanics covered in upper-level undergraduate quantum mechanics courses. The concepts included in the quantum mechanics formalism and postulate survey (QMFPS) focus on Dirac notation, the Hilbert space, state vectors, physical observables and their corresponding Hermitian operators, compatible and incompatible observables, quantum measurement, time dependence of quantum states and expectation values, and spin angular momenta. Here we describe the validation and administration of the survey, which has been administered to over 400 upper-level undergraduate and graduate students from six institutions. The QMFPS is valid and reliable for use as a low-stakes test to measure the effectiveness of instruction in an undergraduate quantum mechanics course that covers relevant content. The survey can also be used by instructors to identify students’ understanding of the formalism and postulates of quantum mechanics at the beginning and end of a graduate quantum mechanics course since graduate students are expected to have taken an undergraduate quantum mechanics course that covers the content included in the survey. We found that undergraduate students who engaged with research-validated learning tools performed better than students who did not on the QMFPS after the first semester of a junior or senior level quantum mechanics course. In addition, the performance of graduate students on QMFPS after instruction in the first semester of a core graduate-level quantum mechanics course was significantly better than the performance of undergraduate students at the end of the first semester of an undergraduate quantum mechanics course. However, both undergraduate and graduate students struggled with many questions on the QMFPS. A comparison with the base line data on the validated QMFPS presented here can aid instructors in assessing the effectiveness of their instructional approaches and help them identify the difficulties their students have with quantum formalism and postulates in order to help students develop a solid grasp of the formalism and postulates of quantum mechanics.

Much of the literature contributing to physics instructors’ knowledge of student ideas (KSI) reports common patterns of reasoning that are framed as discontinuous with canonical concepts. Our work contributes new KSI about mechanical wave propagation from a resources perspective, framing student thinking in terms of context-sensitive pieces of knowledge that are continuous with canonical physics concepts. The intent of this work is to inform instruction on mechanical waves by identifying and illustrating some of the conceptual resources that instructors might expect their students to use. To support instructor predictions about student thinking, we identify resources that are common across multiple samples and questions. Our data include written responses to three versions of a conceptual question about mechanical pulse propagation. We use an emergent coding scheme to characterize a total of 851 written responses from 6 universities in the United States. Our analysis reveals three common conceptual resources: (i) properties of the medium either impede or facilitate the motion of the pulse, (ii) the speed or duration of transverse motion affects pulse speed, and (iii) the speed of the pulse is affected by its kinetic energy. We show how each of these resources can be viewed as continuous with formal understandings of pulse propagation.

Here we demonstrate the benefits of a new curriculum for introductory calculus-based physics that motivates classical mechanics using a modified version of Hamiltonian mechanics. This curriculum shifts the initial focus of instruction away from forces and the associated vector mathematics, which are known to be problematic for students, to the scalar quantity energy, which is more closely aligned with their previously established intuition, and associated differential and integral calculus. We show that implementation of this calculus-enhanced “energy-first” curriculum resulted in higher normalized gains on the Force Concept Inventory exam for all students and improved performance in downstream engineering courses for students with lower ACT math scores. In other words, the downstream benefits were largest for students with lower math abilities who also pose a larger retention risk. This new curriculum thus has the potential to improve student retention by specifically helping the students who need help the most, including traditionally underserved populations who often have weaker mathematics preparation. We propose future work to investigate whether this new curriculum has lowered the math transference barrier to learning in introductory physics, resulting concomitantly in improvements in student learning of classical mechanics and in student fluency with applied mathematics.

An essential skill for success in upper-division physics curricula is the ability to work with and apply mathematical and physical concepts through Cartesian and non-Cartesian coordinate systems. These skills are most notably necessary in electricity and magnetism, wherein students must build and solve integrals and perform vector derivatives in Cartesian, spherical, and cylindrical coordinate systems. While limited in scope, early investigations into students understanding of coordinate systems has shown that students have considerable difficulty. Through the analysis of four one-on-one semistructured interviews, students were observed to activate numerous productive resources, despite frequently incorrect answers. This paper will revisit our previously identified student resource clusters for unit vectors. We also identify additional resources that cast additional light on students’ thinking about position vectors, and velocity vectors in non-Cartesian coordinate systems. Through detailed analysis of a single student, we identify several groupings of resources that will serve as a baseline for future analysis of additional students and provide initial insight into potential curriculum development.

We present an empirical analysis of students’ use of partial derivatives in the context of problem solving in upper-division thermodynamics. Task-based individual interviews were conducted with eight middle-division physics students. The interviews involved finding a partial derivative from information presented in a table and a contour plot. Using thematic analysis, we classified student problem-solving strategies into two principal categories: the analytical derivation strategy and the graphical analysis strategy. We developed a new flowchartlike analysis method: representational transformation. Our analysis of students’ strategies using this method revealed three types of transformation phenomena: translation, consolidation, and dissociation. Students in this study did not seem to have much difficulty with the concepts underlying the partial derivative; instead, they seemed to have difficulty with the transformation phenomena, particularly the consolidation of multiple representations into a single representation.

Special relativity theory (SRT) has recently gained popularity as a first introduction to “modern” physics thinking in upper level secondary physics education. A central idea in SRT is the absolute speed of light, with light propagating with uniform speed relative to the reference frame of the observer. Previous research suggests that students, building on their prior understandings of light propagation and relative motion, develop misunderstandings of this idea. The available research provides little detail on the reasoning processes underlying these misunderstandings. We therefore studied secondary education students’ preinstructional reasoning about the speed of light in a qualitative study, probing students’ reasoning through both verbal reasoning and drawing. Event diagrams (EDs) were used as a representational tool to support student reasoning. Results show that students productively use EDs to reason with light propagation. In line with previous research, we found two alternative reference frames students could use for uniform light propagation. Most students show a flexibility in their use of reference frame: They not only evaluate light propagation in their preferred frame of reference, but also relative to other frames. Some students experienced conflict between an alternative reference frame and the speed of light and changed their reasoning because of that. This finding suggests promising directions for designing education.

Module analysis for multiple-choice responses (MAMCR) was applied to a large sample of Force Concept Inventory (FCI) pretest and post-test responses ($N_{\mathrm{pre}}=4509$ and $N_{\mathrm{post}}=4716$) to replicate the results of the original MAMCR study and to understand the origins of the gender differences reported in a previous study of this dataset. When the results of MAMCR could not be replicated, a modification of the method was introduced, modified module analysis (MMA). MMA was productive in understanding the structure of the incorrect answers in the FCI, identifying 9 groups of incorrect answers on the pretest and 11 groups on the post-test. These groups, in most cases, could be mapped on to common misconceptions used by the authors of the FCI to create distractors for the instrument. Of these incorrect answer groups, 6 of the pretest groups and 8 of the post-test groups were the same for men and women. Two of the male-only pretest groups disappeared with instruction while the third male-only pretest group was identified for both men and women postinstruction. Three of the groups identified for both men and women on the post-test were not present for either on the pretest. The rest of the identified incorrect answer groups did not represent misconceptions, but were rather related to the blocked structure of some FCI items where multiple items are related to a common stem. The groups identified had little relation to the gender unfair items previously identified for this dataset, and therefore, differences in the structure of student misconceptions between men and women cannot explain the gender differences reported for the FCI.

A common hope of many physics educators and researchers is that students leave the course with a stronger sense that physics is relevant to them than when they entered the course. Multiple survey measures have attempted to measure shifts in students’ beliefs on the relevance of physics but frequently the results show a negative shift in students’ beliefs and are often reported as a failure of students to “see the relevance.” We challenge this view by highlighting the limitations of attitudinal and epistemological surveys’ ability to probe relevance. We then articulate a more expansive view of relevance using ecological systems theory that serves as a lens we apply to analyze students’ experiences. Instead of deficit-framing students’ abilities or challenging their beliefs about the relevance of physics, we show how incorporating their rich disciplinary experiences can make physics classrooms truly more relevant.

The use of machine learning and data mining techniques across many disciplines has exploded in recent years with the field of educational data mining growing significantly in the past 15 years. In this study, random forest and logistic regression models were used to construct early warning models of student success in introductory calculus-based mechanics (Physics 1) and electricity and magnetism (Physics 2) courses at a large eastern land-grant university. By combining in-class variables such as homework grades with institutional variables such as cumulative GPA, we can predict if a student will receive less than a “B” in the course with 73% accuracy in Physics 1 and 81% accuracy in Physics 2 with only data available in the first week of class using logistic regression models. The institutional variables were critical for high accuracy in the first four weeks of the semester. In-class variables became more important only after the first in-semester examination was administered. The student’s cumulative college GPA was consistently the most important institutional variable. Homework grade became the most important in-class variable after the first week and consistently increased in importance as the semester progressed; homework grade became more important than cumulative GPA after the first in-semester examination. Demographic variables including gender, race or ethnicity, and first generation status were not important variables for predicting course grade.

Students’ intentions to persevere and their career choices in science, technology, engineering, and math fields can be impacted by their physics identities. Women are severely underrepresented at all levels in physics and engineering. Physics in particular has stereotypes about being a discipline for brilliant men. Therefore, it is particularly difficult for women who do not fit the description of a stereotypical physicist to develop a physics identity. Thus, understanding the factors underlying physics identity in introductory physics classrooms is important for creating an equitable and inclusive physics learning environment and has the potential to at least partly explain the current underrepresentation of women in physics-related majors and careers. In this study, we examined physics identity and several other motivational constructs of male and female students by administering a survey in introductory calculus-based physics courses at a large research university. We found gender differences in how students identify as a physics person and how their perceived recognition from others, such as their teaching assistants or instructors, peers, or family members relates to their physics identities. We tested separate models by gender that examined how different motivational constructs relate to students’ physics identities. We found that the perception of being recognized by influential others such as the course instructor or teaching assistants was differentially related to female and male students’ physics self-efficacy and sense of belonging in the physics classroom. These findings call for improving the physics learning environments to make them equitable so that all students have a high sense of belonging and self-efficacy and opportunity to develop a strong physics identity.

Conceptual inventory surveys are routinely used in education research to identify student learning needs and assess instructional practices. Students might not fully engage with these instruments because of the low stakes attached to them. This paper explores tests that can be used to estimate the percentage of students in a population who might not have taken such surveys seriously. These three seriousness tests are the pattern recognition test, the easy questions test, and the uncommon answers test. These three tests are applied to sets of students who were assessed either by the Force Concept Inventory, the Conceptual Survey of Electricity and Magnetism, or the Brief Electricity and Magnetism Assessment. The results of our investigation are compared to computer simulated populations of random answers.

We examined the variation in Force and Motion Conceptual Evaluation (FMCE) preclass scores according to the type of physics class a student had in high school. This was done for students who had taken Physics 1 at a highly selective university. The majority of students had taken an Advanced Placement (AP) physics course, allowing us to examine in detail the correlation between FMCE score and both taking an AP course and the score on the AP exam. We also carried out regression analyses to determine how FMCE prescores and course final exam scores depend on taking an AP course and AP exam scores, when math SAT scores are included as a proxy of students’ general level of college preparation. Students who take any type of physics course in high school outscore those with no prior physics experience by 26%, but students who have taken AP physics do not significantly outperform those who have taken a normal high school physics course—many students who took AP physics scored very low on the FMCE. Furthermore, while taking AP physics predicts a statistically significant increase in FMCE scores, the effect size (0.30) is smaller than that seen for math SAT scores (0.47). We found that AP exam scores were not correlated with final exam scores after controlling for math SAT score. In all, the data indicate that, though taking an AP physics course is correlated with greater conceptual understanding of physics, scoring well on the AP physics exam may be a rather weak measure of conceptual understanding of physics or the mastery of physics one would expect students to achieve from an introductory university physics course. This work was done with an unusual population. Further studies are needed to determine how general the weak relationship is between AP exam performance and FMCE performance that we observed.

The understanding of graphs and extraction of relevant information from graphs plays a major role in physics education and is also important in several related fields. Recently, Susac et al. [Phys. Rev. Phys. Educ. Res. 14, 020109 (2018)] compared physics and psychology students’ understanding of graphs in the contexts of physics and finance. They showed that physicists scored significantly higher in both domains and that all students solved the slope problems better than the area problems. Moreover, eye-tracking data revealed that physics students spent more time on problems associated with the area under the graph and focused longer on the axis labels of finance graphs, indicating higher cognitive demands. In this eye-tracking study, we aim for a generalization of the results obtained by Susac et al. by comparing physics students to another nonphysics sample, viz., economics students. The findings broadly confirm the results of Susac et al.; that is, physics students perform better than nonphysics students. While economics students likely have better prior knowledge on finance context than psychology students, the physics students still outperform them on the finance questions. In contrast to the work by Susac et al., both groups of students had the same visit duration on the graphs, consequently proving total dwell time to be an inadequate predictor of performance. Instead, we identify that attention on concept-specific areas of interest within the graphs discriminates the correct from the incorrect performers. Furthermore, we analyzed the confidence level of the two student groups and found that physics students have a higher ability to correctly judge their own performance compared to economics students. Overall, our results highlight the importance of an instructional adjustment towards a more mathematical- and graphical-based education.

Understanding the ways in which Black folks identify as physicists can provide useful information about the facets of the physics discipline that perpetuate systems of oppression. In prior work, we developed the critical physics identity (CPI) framework to identify the structures and systems that impact the ways that people identify with the physics discipline. In this paper, we apply the CPI to deconstruct the factors that influence physics identity for a sample of undergraduate, graduate, and professional Black physicists. Using the CPI constructs, we classify important internal and external factors that influence how they identify, or not, as a physics person. We find that racialized resources were more influential than physics identity constructs, and document the subcodes of each CPI construct that are discussed most frequently among participants. We note variations between early and later career physicists and between men and women.

We have studied the impact of incoming preparation and demographic variables on student performance on the final exam in the standard introductory calculus-based mechanics course at three different institutions. Multivariable regression analysis was used to examine the extent to which exam scores can be predicted by a variety of variables that are available to most faculty and departments. The results are surprisingly consistent across the institutions, with only math SAT or ACT scores and concept inventory prescores having predictive power. They explain 20%–30% of the variation in student exam performance in all three cases. In all cases, although there appear to be gaps in exam performance if one considers only demographic variables (gender, underrepresented minority, first generation), once these two proxies of incoming preparation are controlled for, there is no longer a demographic gap. There is only a preparation gap that applies equally across the entire student population. This work shows that to properly understand differences in student performance, it is important to do statistical analyses that take multiple variables into account, covering both subject-specific and general preparation. Course designs and teaching better matched to the incoming student preparation will likely eliminate performance gaps across demographic groups, while also improving the success of all students.

We report on a study investigating the influence of context, direction of translation, and function type on undergraduate students’ ability to translate between graphical and symbolic representations of mathematical relations. Students from an algebra-based and a calculus-based physics course were asked to solve multiple-choice items in which they had to link a graph to an equation or vice versa and explain their answer. The first part of the study focuses on the accuracy of the chosen alternative. Using a generalized estimating of equations (GEE) analysis we find that mathematics items are solved better than physics or kinematics items; that items starting from a graph are solved better than those starting from an equation; and that items on inversely proportional functions are the hardest for students. Quantitatively we see big differences in the number of correct answers between the algebra-based and calculus-based cohorts, but qualitatively the effects of the item variables on the accuracy are the same for both groups. The second part of the study focuses on students’ argumentations for the chosen alternative. We observe that students use their physical knowledge 2 to even 3 times as much in kinematics items than in other physics items. When there is an effect of context on argument use, we observe that the use of an argument in a mathematics context differs significantly from the use of that argument in the physics and/or kinematics context. With regard to function type, students explain their choice of answer in items on inversely proportional functions with different arguments and fail more often to answer correctly. In general, the data also show that students from the calculus-based course use more mathematical arguments and score better on the items.

[This paper is part of the Focused Collection on Quantitative Methods in PER: A Critical Examination.] A relevant aim of research in education is to find and study the reasoning lines that students deploy when dealing with problematic situations. This can be done through an analysis of the answers students give to a questionnaire. In this paper, we discuss some methodological aspects involved in the quantitative analysis of a questionnaire by means of two different clustering methods, a hierarchical one and a nonhierarchical one. We start from the coding procedures needed to obtain analyzable data from the questionnaire and from a definition of a correlation coefficient suitable for measuring student similarity in the case of binary coding of student answers. Then, criteria to choose the optimal number of clusters are discussed, and for the same purpose a new coefficient is introduced that measures the total amount of information we can obtain from a clustering solution. We show that each cluster can be characterized by its centroid that summarizes the answers most frequently given by the cluster students to the questionnaire. Finally, an example of the application of these procedures to a student sample is given, and a comparison between the two clustering methods is discussed.

[This paper is part of the Focused Collection on Quantitative Methods in PER: A Critical Examination.] The Rasch model is a probabilistic model which describes the interaction of persons (test takers or survey respondents) with test or survey items and is governed by two parameters: item difficulty and person ability. Rasch measurement parallels physical measurement processes by constructing and using linear person and item measures that are independent of the particular characteristics of the sample and the test items along a unidimensional construct. The model’s properties make it especially suitable for test construction and evaluation as well as the development and use of surveys. The evaluation of item fit with the model can pinpoint problematic items and flag idiosyncratic respondents. The possibility of determining sample—independent item difficulties makes it possible to use the Rasch model for linking tests and tracking students’ progression. The use of the Rasch model in PER is continuously increasing. We provide an overview and examples of its use and benefits, and also outline common mistakes or misconceptions made by researchers when considering the use of the Rasch model. We focus in particular on the question of how Rasch modeling can improve some common practices in PER, such as test construction, test evaluation, and calculation of student gain on PER diagnostic instruments.

[This paper is part of the Focused Collection on Quantitative Methods in PER: A Critical Examination.] A common goal in discipline-based education research (DBER) is to determine how to improve student outcomes. Linear regression is a common technique used to test hypotheses about the effects of interventions on continuous outcomes (such as exam score) as well as control for student nonequivalence in quasirandom experimental designs. (In quasirandom designs, subjects are not randomly assigned to treatments. For example, when treatment is assigned by classroom, and observations are made on students, the design is quasirandom because treatment is assigned to classroom, not subject (students).) However, many types of outcome data cannot be appropriately analyzed with linear regression. In these instances, researchers must move beyond linear regression and implement alternative regression techniques. For example, student outcomes can be measured on binary scales (e.g., pass or fail), tightly bound scales (e.g., strongly agree to strongly disagree), or nominal scales (i.e., different discrete choices for example multiple tracks within a physics major), each necessitating alternative regression techniques. Here, we review extensions of linear modeling—generalized linear models (glms)—and specifically compare five glms that are useful for analyzing DBER data: logistic, binomial, proportional odds (also called ordinal; including censored regression), multinomial, and Poisson (including negative binomial, hurdle, and zero-inflated) regression. We introduce a diagnostic tool to facilitate a researcher’s identification of the most appropriate glm for their own data. For each model type, we explain when, why, and how to implement the regression approach. When: we provide examples of the types of research questions and outcome data that would motivate this regression approach, including citations to articles in the DBER literature. Why: we name which linear regression assumption is violated by the data type. How: we detail implementation and interpretation of this modeling approach in R, including R syntax and code, and how to discuss the regression output in research papers. Code accompanying each analysis can be found in the online github repository that is associated with this paper (https://github.com/ejtheobald/BeyondLinearRegression). This paper is not an exhaustive review of regression techniques, nor does it review nonregression-based analyses. Rather, it aims to compile and summarize regression techniques useful for the most common types of DBER data and provide examples, citations, and heavily annotated R code so that researchers can easily implement the technique in their work.

[This paper is part of the Focused Collection on Quantitative Methods in PER: A Critical Examination.] The education of physicists requires annual progress in mathematics through grades K–12, continuing on to high school physics, undergraduate instruction, and often many years of postgraduate study. The traditional domain of physics education research is the study of pedagogical techniques for a single lesson or a single course, but the physics and physics education research communities will need to study the complete educational system to increase performance and participation, especially for underrepresented groups. Analysis of long-term student outcomes and policy impacts requires predictive longitudinal techniques. We begin with concepts from fluid and statistical mechanics—trajectories and streamlines—to visualize longitudinal outcomes and make predictions. Drawing on coarse-graining procedures to model the movement of particles in a fluid, the longitudinal data are sorted into score bins to depict the flow of scores through time: trajectories depict the average scores over time for initial score bins and streamlines provide an approximate way to calculate the flow of student scores over many years based on only two consecutive years of data. However, due to the partially stochastic nature of observed scores, the coarse-graining procedure that sorts students into score bins amplifies a statistical phenomenon known as regression to the mean (RTM). As a result, streamlines do not provide an accurate prediction for the future performance of students. Here we discuss a new coarse-graining procedure, regression-corrected (RC) grouping, which reduces RTM in the streamlines. We explain the idea of RC streamlines through a toy model of freshman physics performance and then apply them to the realistic setting of the Texas State Longitudinal Data System, which contains standardized testing data for students throughout primary and secondary school since 2003. We show that the RC streamlines accurately predict trajectories, using 2 or 3 years of data. Therefore RC streamlines can be used to identify the effects of academic interventions on a time scale comparable to that of policy changes. We illustrate this assertion by examining a particular policy intervention, Texas’s Student Success Initiative.